1. Introduction
In 1967, Gardner et al [1] proposed the famous inverse scattering method (ISM) when studying the fast decay initial value problem of the Korteweg–de Vries equation, which is a powerful tool for solving the initial value problem of nonlinear integrable systems. However, because the ISM was only used to discuss the initial value problem of nonlinear integrable equations and the limitation of the initial value conditions is suitable for infinity, how to extend ISM to the initial-boundary value problems (IBVPs) of nonlinear integrable systems is a major challenge for soliton theory research. In 1997, Fokas [2] extended the ISM and proposed a unified transformation method (UTM) to analyze the IBVPs of partial differential equations [3]. In 2008, Lenells [4] used UTM to analyze the IBVPs of the following derivative nonlinear Schrödinger (DNLS) equation [5–7]
$\begin{eqnarray}{\rm{i}}{q}_{t}+{q}_{{xx}}={\rm{i}}{\left(| q{| }^{2}q\right)}_{x}.\end{eqnarray}$
Equation (1.1 ) has an important application in plasma physics, which is a model for Alfvén waves propagating parallel to the ambient magnetic field [8, 9]. Since then, more and more mathematical physicists have paid attention to the UTM to study the IBVPs of integrable equations [10–18]. In 2012, Lenells extended UTM to integrable systems related to high-matrix spectral [19], and used UTM to analyze the IBVPs of the Degasperis–Procesi equation [20, 21]. In 2013, Xu and Fan discussed the IBVPs of the Sasa–Satsuma equation through UTM [22], and gave the proof of the existence and uniqueness of the solution of the IBVPs of the integrable equation with higher-order matrix spectrum through analyzing a three-wave equation [23]. Subsequently, more and more scholars have studied the IBVPs of integrable equations with higher-order matrix spectral [24–27]. Particularly, the soliton solutions and the long-time asymptotic behavior for the integrable models can be solved by constructing a Riemann–Hilbert (RH) problem. Such as, Wang and Wang investigated the long-time asymptotic behavior of the Kundu–Eckhaus equation [28]. Yang and Chen obtained the high-order soliton matrix form solution of the Sasa–Satsuma equation [29]. Ma analyzed multicomponent AKNS integrable hierarchies [30], etc.
In 1987, Clarkson and Cosgrove [31] proposed a generalized derivative NLS (GDNLS) equation in the form of1.2 ) has several applications in optical fibers, nonlinear optics, weakly nonlinear dispersion water waves, quantum field theory, and plasma physics [32, 33], etc. As an example, equation (1.2 ) can be used to simulate single-mode propagation in the optical fibers, which enjoys traveling and stationary kink envelope solutions of monotonic and oscillatory type. However, it is well know that equation (1.2 ) has Painlevé property only if $\kappa =\tfrac{1}{4}\beta (2\beta -\alpha )$ holds. At this time, equation (1.2 ) is reduced to an integrable GDNLS model as follows1.3 ) becomes to the DNLS-I (Kaup–Newell) equation (1.1 ), and if α ≠ 0, β = 0, the equation (1.3 ) becomes to the DNLS-II (Chen–Lee–Liu) equation1.3 ) have been discussed [35]. However, as far as we know, the IBVPs of equation (1.3 ) have not been analyzed. So we will utilize UTM to study the IBVPs of equation (1.3 ) on the half-line domain Γ = {(x, t): 0 < x < ∞, 0 < t < T} here. Similar to DNLS equation [18] on the interval, the IBVPs of equation (1.3 ) on the interval will be studied in our future paper.
$\begin{eqnarray}{\rm{i}}{q}_{t}={q}_{{xx}}+{\rm{i}}\alpha | q{| }^{2}{q}_{x}+{\rm{i}}\beta {q}^{2}{\bar{q}}_{x}+\kappa | q{| }^{4}q,\alpha \ne \beta ,\end{eqnarray}$
where q is the amplitude of the complex field envelope. The equation ( $\begin{eqnarray}{\rm{i}}{q}_{t}={q}_{{xx}}+{\rm{i}}\alpha | q{| }^{2}{q}_{x}+{\rm{i}}\beta {q}^{2}{\bar{q}}_{x}+\displaystyle \frac{1}{4}\beta (2\beta -\alpha )| q{| }^{4}q,\alpha \ne \beta .\end{eqnarray}$
Given α = 2β ≠ 0, the equation ( $\begin{eqnarray}{\rm{i}}{q}_{t}={q}_{{xx}}+{\rm{i}}\alpha | q{| }^{2}{q}_{x},\end{eqnarray}$
whose IBVPs on the half-line has been solved [34]. Recently, the conservation laws of equation (The design structure of this paper is as follows. In section 2 , we give spectral analysis of the Lax pair of equation (1.3 ). In section 3 , some key functions f(η), s(η), F(η), S(η) are further analyzed. In section 4 , the RH problem is proposed. Finally, some conclusions and discussions are given in section 5 .
2. The spectral analysis
The GDNLS equation (1.3 ) enjoys a Lax pair as follows [35]
$\begin{eqnarray}{{\rm{\Phi }}}_{x}=U(x,t,\eta ){\rm{\Phi }},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Phi }}}_{t}=V(x,t,\eta ){\rm{\Phi }},\end{eqnarray}$
where Φ = (Φ1, Φ2)T is the vector eigenfunction, the 2 × 2 matrices U(x, t, η), V(x, t, η) are given by the following form $\begin{eqnarray}\begin{array}{rcl}U(x,t,\eta ) & = & \left(\begin{array}{cc}\tfrac{-{\rm{i}}{\eta }^{2}}{\alpha -\beta }-\tfrac{{\rm{i}}}{4}(\alpha -2\beta )| q{| }^{2} & \eta \bar{q}\\ \eta q & -\tfrac{-{\rm{i}}{\eta }^{2}}{\alpha -\beta }+\tfrac{{\rm{i}}}{4}(\alpha -2\beta )| q{| }^{2}\end{array}\right)\\ & = & \displaystyle \frac{-{\rm{i}}{\eta }^{2}}{\alpha -\beta }{\sigma }_{3}+\eta Q-\displaystyle \frac{{\rm{i}}}{4}(\alpha -2\beta ){Q}^{2}{\sigma }_{3},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}V(x,t,\eta )=\left(\begin{array}{cc}{V}_{11} & \tfrac{2}{\alpha -\beta }{\eta }^{3}\bar{q}+\tfrac{\alpha }{2}| q{| }^{2}\bar{q}-{\rm{i}}{\bar{q}}_{x}\\ \tfrac{2}{\alpha -\beta }{\eta }^{3}q+\tfrac{\alpha }{2}| q{| }^{2}q-{\rm{i}}{q}_{x} & -{V}_{11}\end{array}\right)\\ =\,\displaystyle \frac{2{\rm{i}}{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}{\sigma }_{3}-[{\rm{i}}{\eta }^{2}{Q}^{2}+\displaystyle \frac{{\rm{i}}}{8}({\alpha }^{2}-\alpha \beta -2{\beta }^{2}){Q}^{4}\\ \ +\,\displaystyle \frac{\alpha -2\beta }{4}({{QQ}}_{x}-{Q}_{x}Q)]{\sigma }_{3}\\ \ +\,\displaystyle \frac{2}{\alpha -\beta }{\eta }^{3}Q+\displaystyle \frac{\alpha }{2}{Q}^{3}-{\rm{i}}{Q}_{x}{\sigma }_{3},\end{array}\end{eqnarray}$
with ${V}_{11}=-\tfrac{2{\rm{i}}{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}-{\rm{i}}{\eta }^{2}| q{| }^{2}$ $+\,\tfrac{{\rm{i}}}{8}({\alpha }^{2}-\alpha \beta -2{\beta }^{2})| q{| }^{4}\,-\tfrac{\alpha -2\beta }{4}(\bar{q}{q}_{x}-{\bar{q}}_{x}q)$ and $\begin{eqnarray}\begin{array}{l}{\sigma }_{3}=\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right),\ \ Q=\left(\begin{array}{cc}0 & \bar{q}\\ q & 0\end{array}\right).\end{array}\end{eqnarray}$
2.1. The exact one-form
The equations (2.1a ), (2.1b ) is equivalent to
$\begin{eqnarray}{{\rm{\Phi }}}_{x}+\displaystyle \frac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}{\sigma }_{3}{\rm{\Phi }}=M{\rm{\Phi }},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Phi }}}_{t}+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}{\sigma }_{3}{\rm{\Phi }}=N{\rm{\Phi }},\end{eqnarray}$
where α ≠ β, complex number η is a spectral parameter and $\begin{eqnarray*}\begin{array}{rcl}M & = & \eta Q-\displaystyle \frac{{\rm{i}}}{4}(\alpha -2\beta ){Q}^{2}{\sigma }_{3},\\ N & = & -[{\rm{i}}{\eta }^{2}{Q}^{2}+\displaystyle \frac{{\rm{i}}}{8}({\alpha }^{2}-\alpha \beta -2{\beta }^{2}){Q}^{4}\\ & & +\displaystyle \frac{\alpha -2\beta }{4}({{QQ}}_{x}-{Q}_{x}Q)]{\sigma }_{3}\\ & & +\displaystyle \frac{2}{\alpha -\beta }{\eta }^{3}Q+\displaystyle \frac{\alpha }{2}{Q}^{3}-{\rm{i}}{Q}_{x}{\sigma }_{3}.\end{array}\end{eqnarray*}$
One can introduce $\Psi$(x, t, η) by2.4a ), (2.4b ) become to
$\begin{eqnarray}{\rm{\Psi }}(x,t,\eta )={\rm{\Phi }}(x,t,\eta ){{\rm{e}}}^{[\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t]{\sigma }_{3}},\end{eqnarray}$
hence, equations ( $\begin{eqnarray}{{\rm{\Psi }}}_{x}+\displaystyle \frac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}[{\sigma }_{3},{\rm{\Psi }}]=M{\rm{\Psi }},\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Psi }}}_{t}+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}[{\sigma }_{3},{\rm{\Psi }}]=N{\rm{\Psi }},\end{eqnarray}$
where [σ3, $\Psi$] = σ3$\Psi$ − $\Psi$σ3, it is easy to see that the above equations give the following full differential $\begin{eqnarray}\begin{array}{l}{\rm{d}}({{\rm{e}}}^{[\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t]{\hat{\sigma }}_{3}}{\rm{\Psi }}(x,t,\eta ))\\ =\,{{\rm{e}}}^{[\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t]{\hat{\sigma }}_{3}}(M{\rm{d}}x+N{\rm{d}}t){\rm{\Phi }}(x,t,\eta ),\end{array}\end{eqnarray}$
where ${\hat{\sigma }}_{3}$ is a matrix operator (see [17]).One supposes that the following asymptotic expansion2.6a ), (2.6b ). Substituting equation (2.8 ) into equation (2.6a ) and comparing the coefficients for ηj, one can get
$\begin{eqnarray}{\rm{\Psi }}(x,t,\eta )={D}_{0}+\displaystyle \frac{{D}_{1}}{\eta }+\displaystyle \frac{{D}_{2}}{{\eta }^{2}}+\displaystyle \frac{{D}_{3}}{{\eta }^{3}}+O\left(\displaystyle \frac{1}{{\eta }^{4}}\right),\eta \to \infty ,\end{eqnarray}$
is a solution of equations ( $\begin{eqnarray}\begin{array}{l}O({\eta }^{2})\ :\displaystyle \frac{{\rm{i}}}{\alpha -\beta }[{\sigma }_{3},{D}_{0}]=0,\\ O({\eta }^{1})\ :\displaystyle \frac{{\rm{i}}}{\alpha -\beta }[{\sigma }_{3},{D}_{1}]={{QD}}_{0},\\ O({\eta }^{0})\ :{D}_{0x}+\displaystyle \frac{{\rm{i}}}{\alpha -\beta }[{\sigma }_{3},{D}_{2}]\\ =\,{{QD}}_{1}-\displaystyle \frac{{\rm{i}}(\alpha -2\beta )}{4}{Q}^{2}{\sigma }_{3}{D}_{0}.\end{array}\end{eqnarray}$
From O(η2), one finds that D0 enjoys a diagonal matrix form denoted as2.8 ) into the equation (2.6b ) and comparing the coefficient for η, we get
$\begin{eqnarray*}{D}_{0}=\left(\begin{array}{cc}{D}_{0}^{11} & 0\\ 0 & {D}_{0}^{22}\end{array}\right).\end{eqnarray*}$
From O(η1), one obtains $\begin{eqnarray}{D}_{1}^{({\rm{od}})}=\displaystyle \frac{{\rm{i}}}{2}(\alpha -\beta )\left(\begin{array}{cc}0 & -\bar{q}{D}_{0}^{22}\\ {{qD}}_{0}^{11} & 0\end{array}\right),{D}_{0x}=\displaystyle \frac{{\rm{i}}}{4}\alpha | q{| }^{2}{\sigma }_{3}{D}_{0},\end{eqnarray}$
where ${D}_{1}^{({\rm{od}})}$ denotes the off-diagonal part of D1. At the same time, substituting equation ( $\begin{eqnarray}\begin{array}{l}O({\eta }^{4})\ :\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}[{\sigma }_{3},{D}_{0}]=0,\\ O({\eta }^{3})\ :\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}[{\sigma }_{3},{D}_{1}]=\displaystyle \frac{2}{\alpha -\beta }{{QD}}_{0},\\ O({\eta }^{2})\ :\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}[{\sigma }_{3},{D}_{2}]=-{\rm{i}}{Q}^{2}{\sigma }_{3}{D}_{0}+\displaystyle \frac{2}{\alpha -\beta }{{QD}}_{1},\\ O({\eta }^{1})\ :\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}[{\sigma }_{3},{D}_{3}]=-{\rm{i}}{Q}^{2}{\sigma }_{3}{D}_{1}+\displaystyle \frac{2}{\alpha -\beta }{{QD}}_{2},\\ O({\eta }^{0})\ :{D}_{0t}+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}[{\sigma }_{3},{D}_{4}]\\ =\,-{\rm{i}}{Q}^{2}{\sigma }_{3}{D}_{2}+\displaystyle \frac{2}{\alpha -\beta }{{QD}}_{3}\\ -\,\left[\displaystyle \frac{{\rm{i}}}{8}({\alpha }^{2}-\alpha \beta -2{\beta }^{2}){Q}^{4}+\displaystyle \frac{\alpha -2\beta }{4}({{QQ}}_{x}-{Q}_{x}Q)\right]{\sigma }_{3}{D}_{0}\\ +\,\left(\displaystyle \frac{\alpha }{2}{Q}^{3}-{\rm{i}}{Q}_{x}{\sigma }_{3}\right){D}_{0}.\end{array}\end{eqnarray}$
Through tedious calculation, one gets2.1a ), (2.1b ) admit the following conservation law2.10 ) and (2.12 ) for D0 are consistent, then, one defines2.13 ) is independent of the integration path and Ω is independent of η, one can introduce a key function G(x, t, η) by2.7 ) is equal to
$\begin{eqnarray}{D}_{0t}=\left[\displaystyle \frac{{\rm{i}}}{8}({\alpha }^{2}+\alpha \beta -{\beta }^{2})| q{| }^{4}+\displaystyle \frac{\alpha }{4}(\bar{q}{q}_{x}-q{\bar{q}}_{x})\right]{\sigma }_{3}{D}_{0},\end{eqnarray}$
since equations ( $\begin{eqnarray*}{\left(\displaystyle \frac{{\rm{i}}}{4}\alpha | q{| }^{2}\right)}_{t}={\left[\displaystyle \frac{{\rm{i}}}{8}({\alpha }^{2}+\alpha \beta -{\beta }^{2})| q{| }^{4}+\displaystyle \frac{1}{4}\alpha (\bar{q}{q}_{x}-q{\bar{q}}_{x})\right]}_{x},\end{eqnarray*}$
the equations ( $\begin{eqnarray}{D}_{0}(x,t)={{\rm{e}}}^{{\rm{i}}{\int }_{(\infty ,0)}^{(x,t)}{\rm{\Omega }}(x,t){\sigma }_{3}},\end{eqnarray}$
where Ω is the closed one-form and given by $\begin{eqnarray}\begin{array}{rcl}{\rm{\Omega }}(x,t) & = & {{\rm{\Omega }}}_{1}{\rm{d}}x+{{\rm{\Omega }}}_{2}{\rm{d}}t\\ & = & \displaystyle \frac{1}{4}\alpha | q{| }^{2}{\rm{d}}x+\left[\displaystyle \frac{1}{8}({\alpha }^{2}+\alpha \beta -{\beta }^{2})| q{| }^{4}\right.\\ & & \left.-\displaystyle \frac{{\rm{i}}}{4}\alpha (\bar{q}{q}_{x}-q{\bar{q}}_{x})\right]{\rm{d}}t.\end{array}\end{eqnarray}$
Since the integration of equation ( $\begin{eqnarray}{\rm{\Psi }}(x,t,\eta )={{\rm{e}}}^{{\rm{i}}{\int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}{\hat{\sigma }}_{3}}G(x,t,\eta ){D}_{0}(x,t),\end{eqnarray}$
then, equation ( $\begin{eqnarray}{\rm{d}}({{\rm{e}}}^{(\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t){\hat{\sigma }}_{3}}G(x,t,\eta ))=A(x,t,\eta ),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}A(x,t,\eta ) & = & {{\rm{e}}}^{(\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t){\hat{\sigma }}_{3}}B(x,t,\eta )G(x,t,\eta ),\\ B(x,t,\eta ) & = & {M}_{1}(x,t,\eta ){\rm{d}}x+{N}_{1}(x,t,\eta ){\rm{d}}t\\ & = & {{\rm{e}}}^{-{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}{\hat{\sigma }}_{3}}(M{\rm{d}}x+N{\rm{d}}t-{\rm{i}}{\rm{\Omega }}{\sigma }_{3}).\end{array}\end{eqnarray}$
It follows from M(x, t, η), N(x, t, η) and Ω that2.16 ) becomes to
$\begin{eqnarray}{M}_{1}(x,t,\eta )=\left(\begin{array}{cc}-\displaystyle \frac{{\rm{i}}}{2}(\alpha -\beta )| q{| }^{2} & \eta \bar{q}{{\rm{e}}}^{2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}}\\ \eta q{{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}} & \displaystyle \frac{{\rm{i}}}{2}(\alpha -\beta )| q{| }^{2}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{N}_{1}(x,t,\eta )=\left(\begin{array}{cc}{N}_{11}^{(1)}(x,t,\eta ) & {N}_{12}^{(1)}(x,t,\eta )\\ {N}_{21}^{(1)}(x,t,\eta ) & -{N}_{11}^{(1)}(x,t,\eta )\end{array}\right),\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{rcl}{N}_{11}^{(1)}(x,t,\eta ) & = & -{\rm{i}}{\eta }^{2}| q{| }^{2}-\displaystyle \frac{{\rm{i}}}{8}(2\alpha \beta +{\beta }^{2})| q{| }^{4}\\ & & -\displaystyle \frac{\alpha -\beta }{2}(\bar{q}{q}_{x}-{\bar{q}}_{x}q),\\ {N}_{12}^{(1)}(x,t,\eta ) & = & \left(\displaystyle \frac{2}{\alpha -\beta }{\eta }^{3}\bar{q}+\displaystyle \frac{\alpha }{2}| q{| }^{2}\bar{q}-{\rm{i}}{\bar{q}}_{x}\right){{\rm{e}}}^{2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}},\\ {N}_{21}^{(1)}(x,t,\eta ) & = & \left(\displaystyle \frac{2}{\alpha -\beta }{\eta }^{3}q+\displaystyle \frac{\alpha }{2}| q{| }^{2}q-{\rm{i}}{q}_{x}\right){{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}},\end{array}\end{eqnarray*}$
then equation ( $\begin{eqnarray}{G}_{x}+\displaystyle \frac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}[{\sigma }_{3},G]={M}_{1}G,\end{eqnarray}$
$\begin{eqnarray}{G}_{t}+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}[{\sigma }_{3},G]={N}_{1}G.\end{eqnarray}$
2.2. The three important functions ${\{{G}_{j}(x,t,\eta )\}}_{1}^{3}$
For (x, t) ∈ Γ, we suppose that $q(x,t)\in {\mathbb{S}}$, one defines three eigenfunctions ${\{{G}_{j}(x,t,\eta )\}}_{1}^{3}$ of equations (2.19a ), (2.19b ) given by2.17 ), just replacing G(ξ, τ, η) with Gj(ξ, τ, η), the integral path (xj, tj) → (x, t) is a directed smooth curve and (x1, t1) = (0, 0), (x2, t2) = (0, T), (x3, t3) = ( ∞ , t). Since the integral of equation (2.20 ) has nothing to do with the integral path, we select a special integral path parallel to the coordinate axis as shown in figure 1, then we have
$\begin{eqnarray}{G}_{j}(x,t,\eta )={\boldsymbol{I}}+{\int }_{({x}_{j},{t}_{j})}^{(x,t)}{{\rm{e}}}^{-\left[\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t\right]{\hat{\sigma }}_{3}}{A}_{j}(\xi ,\tau ,\eta ),\end{eqnarray}$
where I = diag{1, 1} is a 2 × 2 unit matrix, Aj(ξ, τ, η) is given by equation ( $\begin{eqnarray}\begin{array}{l}{G}_{1}(x,t,\eta )={\boldsymbol{I}}+{\displaystyle \int }_{0}^{x}{{\rm{e}}}^{-\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}(x-\xi ){\hat{\sigma }}_{3}}({M}_{1}{G}_{1})(\xi ,t,\eta ){\rm{d}}\xi \\ \quad +\,{{\rm{e}}}^{-\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x{\hat{\sigma }}_{3}}{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{-\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(t-\tau ){\hat{\sigma }}_{3}}({N}_{1}{G}_{1})(0,\tau ,\eta ){\rm{d}}\tau ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{G}_{2}(x,t,\eta )={\boldsymbol{I}}+{\displaystyle \int }_{0}^{x}{{\rm{e}}}^{-\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}(x-\xi ){\hat{\sigma }}_{3}}({M}_{1}{G}_{2})(\xi ,t,\eta ){\rm{d}}\xi \\ \quad -\,{{\rm{e}}}^{-\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x{\hat{\sigma }}_{3}}{\displaystyle \int }_{t}^{T}{{\rm{e}}}^{-\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(t-\tau ){\hat{\sigma }}_{3}}({N}_{1}{G}_{2})(0,\tau ,\eta ){\rm{d}}\tau ,\end{array}\end{eqnarray}$
$\begin{eqnarray}{G}_{3}(x,t,\eta )={\boldsymbol{I}}-{\int }_{x}^{\infty }{{\rm{e}}}^{-\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}(x-\xi ){\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,t,\eta ){\rm{d}}\xi .\end{eqnarray}$
Figure 1. The three contours γ1, γ2, γ3 in the (x, t)-domain. |
The first column of equation (2.20 ) enjoys exp $\left[\tfrac{2{\rm{i}}}{\alpha -\beta }{\eta }^{2}(x-\xi )+\tfrac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(t-\tau )\right]$, and the following inequalities
$\begin{eqnarray}{\gamma }_{1}\,:x-\xi \geqslant 0,t-\tau \geqslant 0,\end{eqnarray}$
$\begin{eqnarray}{\gamma }_{2}\,:x-\xi \geqslant 0,t-\tau \leqslant 0,\end{eqnarray}$
$\begin{eqnarray}{\gamma }_{3}:x-\xi \leqslant 0,\end{eqnarray}$
are ture on curves ${\{{\gamma }_{j}\}}_{1}^{3}$, then, the bounded analysis area of eigenfunctions ${\{{G}_{j}(x,t,\eta )\}}_{1}^{3}$ is as follows $\begin{eqnarray}{[{G}_{1}]}_{1}(x,t,\eta )\,:\left\{\mathrm{Im}\displaystyle \frac{{\eta }^{2}}{\alpha -\beta }\geqslant 0\right\}\cap \left\{\mathrm{Im}\displaystyle \frac{{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}\geqslant 0\right\},\end{eqnarray}$
$\begin{eqnarray}{[{G}_{2}]}_{1}(x,t,\eta )\,:\left\{\mathrm{Im}\displaystyle \frac{{\eta }^{2}}{\alpha -\beta }\geqslant 0\right\}\cap \left\{\mathrm{Im}\displaystyle \frac{{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}\leqslant 0\right\},\end{eqnarray}$
$\begin{eqnarray}{[{G}_{3}]}_{1}(x,t,\eta )\,:\left\{\mathrm{Im}\displaystyle \frac{{\eta }^{2}}{\alpha -\beta }\leqslant 0\right\}.\end{eqnarray}$
On the other hand, the second column of equation (2.20 ) contains opposite index terms exp $\left[-\tfrac{2{\rm{i}}}{\alpha -\beta }{\eta }^{2}(x-\xi )-\tfrac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(t-\tau )\right]$.
$\begin{eqnarray}{[{G}_{1}]}_{1}(x,t,\eta )\,:\left\{\mathrm{Im}\displaystyle \frac{{\eta }^{2}}{\alpha -\beta }\leqslant 0\}\cap \{\mathrm{Im}\displaystyle \frac{{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}\leqslant 0\right\},\end{eqnarray}$
$\begin{eqnarray}{[{G}_{2}]}_{1}(x,t,\eta )\,:\left\{\mathrm{Im}\displaystyle \frac{{\eta }^{2}}{\alpha -\beta }\leqslant 0\}\cap \{\mathrm{Im}\displaystyle \frac{{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}\geqslant 0\right\},\end{eqnarray}$
$\begin{eqnarray}{[{G}_{3}]}_{1}(x,t,\eta )\,:\left\{\mathrm{Im}\displaystyle \frac{{\eta }^{2}}{\alpha -\beta }\geqslant 0\right\}.\end{eqnarray}$
Consequently, if we remember that ${\left[{G}_{j}\right]}_{k}(x,t,\eta ),k\,=1,2$ represent k-column of Gj(x, t, η), one can get $\begin{eqnarray}\begin{array}{l}{\rm{for}}\,\alpha \gt \beta ,\\ \left\{\begin{array}{l}{G}_{1}(x,t,\eta )=({\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta ),{\left[{G}_{1}\right]}_{2}^{{L}_{3}}(x,t,\eta )),\\ {G}_{2}(x,t,\eta )=({\left[{G}_{2}\right]}_{1}^{{L}_{2}}(x,t,\eta ),{\left[{G}_{2}\right]}_{2}^{{L}_{4}}(x,t,\eta )),\\ {G}_{3}(x,t,\eta )=({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\\ {\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )),\end{array}\right.\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{\rm{for}}\,\alpha \lt \beta ,\\ \left\{\begin{array}{l}{G}_{1}(x,t,\eta )=({\left[{G}_{1}\right]}_{1}^{{L}_{4}}(x,t,\eta ),{\left[{G}_{1}\right]}_{2}^{{L}_{2}}(x,t,\eta )),\\ {G}_{2}(x,t,\eta )=({\left[{G}_{2}\right]}_{1}^{{L}_{3}}(x,t,\eta ),{\left[{G}_{2}\right]}_{2}^{{L}_{1}}(x,t,\eta )),\\ {G}_{3}(x,t,\eta )=({\left[{G}_{3}\right]}_{1}^{{L}_{1}\cup {L}_{2}}(x,t,\eta ),\\ {\left[{G}_{3}\right]}_{2}^{{L}_{3}\cup {L}_{4}}(x,t,\eta )),\end{array}\right.\end{array}\end{eqnarray}$
where ${G}_{j}^{{L}_{i}}$ represents that the bounded analytic region of ${\{{G}_{j}\}}_{1}^{3}$ is Li, i = 1, 2, 3, 4, and Li are shown in figure 2.
Figure 2. The areas Li, i = 1,…,4 division on the complex η-plane. |
To construct the RH problem of GDNLS equation (1.3 ), we must define another two important special functions $\Psi$(η) and φ(η) by2.27a ) and (2.27b ) we can get
$\begin{eqnarray}{G}_{3}(x,t,\eta )={G}_{1}(x,t,\eta ){{\rm{e}}}^{-[\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t]{\hat{\sigma }}_{3}}\psi (\eta ),\end{eqnarray}$
$\begin{eqnarray}{G}_{2}(x,t,\eta )={G}_{1}(x,t,\eta ){{\rm{e}}}^{-[\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t]){\hat{\sigma }}_{3}}\phi (\eta ),\end{eqnarray}$
upon evaluation at (x, t) = (0, 0) and (x, t) = (0, T), respectively, from equations ( $\begin{eqnarray}{\phi }^{-1}(\eta )={{\rm{e}}}^{-\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}T{\hat{\sigma }}_{3}}{G}_{1}(0,T,\eta ),\psi (\eta )={G}_{3}(0,0,\eta ).\end{eqnarray}$
It follows from (2.27a ), (2.27b ) and equation (2.28 ) that
$\begin{eqnarray}{G}_{2}(x,t,\eta )={G}_{3}(x,t,\eta ){{\rm{e}}}^{-[\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x+\displaystyle \frac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t]{\hat{\sigma }}_{3}}{\left(\psi (\eta )\right)}^{-1}\phi (\eta ).\end{eqnarray}$
Particularly, one also obtains G1(x, t, η), G2(x, t, η) at x = 0
$\begin{eqnarray}\begin{array}{rcl}{G}_{1}(0,t,\eta ) & = & ({\left[{G}_{1}\right]}_{1}^{{L}_{1}\cup {L}_{4}}(0,t,\eta ),{\left[{G}_{1}\right]}_{2}^{{L}_{2}\cup {L}_{3}}(0,t,\eta ))\\ & = & {\boldsymbol{I}}+{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{-\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(t-\tau ){\hat{\sigma }}_{3}}({N}_{1}{G}_{1})(0,\tau ,\eta ){\rm{d}}\tau ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{G}_{2}(0,t,\eta ) & = & ({\left[{G}_{2}\right]}_{1}^{{L}_{2}\cup {L}_{3}}(0,t,\eta ),{\left[{G}_{2}\right]}_{2}^{{L}_{1}\cup {L}_{4}}(0,t,\eta ))\\ & = & {\boldsymbol{I}}-{\displaystyle \int }_{t}^{T}{{\rm{e}}}^{-\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(t-\tau ){\hat{\sigma }}_{3}}({N}_{1}{G}_{2})(0,\tau ,\eta ){\rm{d}}\tau ,\end{array}\end{eqnarray}$
and G1(x, t, η), G3(x, t, η) at t = 0 $\begin{eqnarray}\begin{array}{l}{G}_{1}(x,0,\eta )\\ =\,\left\{\begin{array}{l}({\left[{G}_{1}\right]}_{1}^{{L}_{1}\cup {L}_{2}}(x,0,\eta ),{\left[{G}_{1}\right]}_{2}^{{L}_{3}\cup {L}_{4}}(x,0,\eta ))\,{\rm{for}}\,\alpha \gt \beta ,\\ ({\left[{G}_{1}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,0,\eta ),{\left[{G}_{1}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,0,\eta ))\,{\rm{for}}\,\alpha \lt \beta ,\end{array}\right.\\ =\,{\boldsymbol{I}}+{\int }_{0}^{x}{{\rm{e}}}^{-\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}(x-\xi ){\hat{\sigma }}_{3}}({M}_{1}{G}_{1})(\xi ,0,\eta ){\rm{d}}\xi ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{G}_{3}(x,0,\eta )\\ =\,\left\{\begin{array}{l}({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(z,0,\eta ),{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,0,\eta ))\,{\rm{for}}\,\alpha \gt \beta ,\\ ({\left[{G}_{3}\right]}_{1}^{{L}_{1}\cup {L}_{2}}(z,0,\eta ),{\left[{G}_{3}\right]}_{2}^{{L}_{3}\cup {L}_{4}}(x,0,\eta ))\,{\rm{for}}\,\alpha \lt \beta ,\end{array}\right.\\ =\,{\boldsymbol{I}}-{\int }_{x}^{\infty }{{\rm{e}}}^{-\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}(x-\xi ){\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,0,\eta ){\rm{d}}\xi .\end{array}\end{eqnarray}$
Assume that u0(x) = q(x, t = 0), v0(t) = q(x = 0, t), v1(t) = qx(x = 0, t) are initial condition and boundary conditions of q(x, t) and qx(x, t), then, one get
$\begin{eqnarray}{M}_{1}(x,0,\eta )=\left(\begin{array}{cc}-\displaystyle \frac{{\rm{i}}}{2}(\alpha -\beta )| {u}_{0}{| }^{2} & \eta {\bar{u}}_{0}{{\rm{e}}}^{\tfrac{{\rm{i}}}{2}\alpha {\displaystyle \int }_{0}^{x}| {u}_{0}{| }^{2}{\rm{d}}x}\\ \eta {u}_{0}{{\rm{e}}}^{-\tfrac{{\rm{i}}}{2}\alpha {\displaystyle \int }_{0}^{x}| {u}_{0}{| }^{2}{\rm{d}}x} & \displaystyle \frac{{\rm{i}}}{2}(\alpha -\beta )| {u}_{0}{| }^{2}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{N}_{1}(0,t,\eta )=\left(\begin{array}{cc}{N}_{11}^{(1)}(0,t,\eta ) & {N}_{12}^{(1)}(0,t,\eta )\\ {N}_{21}^{(1)}(0,t,\eta ) & -{N}_{11}^{(1)}(0,t,\eta )\end{array}\right),\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{rcl}{N}_{11}^{(1)}(0,t,\eta ) & = & -{\rm{i}}{\eta }^{2}| {v}_{0}{| }^{2}-\displaystyle \frac{{\rm{i}}}{8}(2\alpha \beta +{\beta }^{2})| {v}_{0}{| }^{4}\\ & & -\displaystyle \frac{\alpha -\beta }{2}({\bar{v}}_{0}{v}_{1}-{\bar{v}}_{1}{v}_{0}),\\ {N}_{12}^{(1)}(0,t,\eta ) & = & \left(\displaystyle \frac{2}{\alpha -\beta }{\eta }^{3}{\bar{v}}_{0}+\displaystyle \frac{\alpha }{2}| {v}_{0}{| }^{2}{\bar{v}}_{0}-{\rm{i}}{\bar{v}}_{1}\right)\\ & & \times {{\rm{e}}}^{{\rm{i}}{\displaystyle \int }_{0}^{t}\left(\displaystyle \frac{1}{4}({\alpha }^{2}+\alpha \beta -{\beta }^{2})| {v}_{0}{| }^{4}-\displaystyle \frac{{\rm{i}}\alpha }{4}({\bar{v}}_{0}{v}_{1}-{v}_{0}{\bar{v}}_{1})\right){\rm{d}}t},\\ {N}_{21}^{(1)}(0,t,\eta ) & = & \left(\displaystyle \frac{2}{\alpha -\beta }{\eta }^{3}{v}_{0}+\displaystyle \frac{\alpha }{2}| {v}_{0}{| }^{2}{v}_{0}-{\rm{i}}{v}_{1}\right)\\ & & \times {{\rm{e}}}^{-{\rm{i}}{\displaystyle \int }_{0}^{t}\left(\displaystyle \frac{1}{4}({\alpha }^{2}+\alpha \beta -{\beta }^{2})| {v}_{0}{| }^{4}-\displaystyle \frac{{\rm{i}}\alpha }{4}({\bar{v}}_{0}{v}_{1}-{v}_{0}{\bar{v}}_{1})\right){\rm{d}}t}.\end{array}\end{eqnarray*}$
2.3. The other properties of the eigenfunctions
The functions
$\begin{eqnarray*}{G}_{j}(x,t,\eta )=({\left[{G}_{j}\right]}_{1}(x,t,\eta ),{\left[{G}_{j}\right]}_{2}(x,t,\eta )),j=1,2,3,\end{eqnarray*}$
enjoy properties as follows| • | det Gj(x, t, η) = 1, j = 1,2,3, |
| • | ${\left[{G}_{1}\right]}_{1}\,{is}\,{analytic}\,{for}$ $\left\{\begin{array}{l}\eta \in {L}_{1},\,{and}\,{continues}\,{to}\,{\bar{L}}_{1},\,\alpha \gt \beta ,\\ \eta \in {L}_{4},\,{and}\,{continues}\,{to}\,{\bar{L}}_{4},\,\alpha \lt \beta ,\end{array}\right.$ |
| • | ${\left[{G}_{1}\right]}_{2}\,{is}\,{analytic}\,{for}$ $\left\{\begin{array}{l}\eta \in {L}_{3},\,{and}\,{continues}\,{to}\,{\bar{L}}_{3},\,\alpha \gt \beta ,\\ \eta \in {L}_{2},\,{and}\,{continues}\,{to}\,{\bar{L}}_{2},\,\alpha \lt \beta ,\end{array}\right.$ |
| • | ${\left[{G}_{2}\right]}_{1}\,{is}\,{analytic}\,{for}$ $\left\{\begin{array}{l}\eta \in {L}_{2},\,{and}\,{continues}\,{to}\,{\bar{L}}_{2},\,\alpha \gt \beta ,\\ \eta \in {L}_{3},\,{and}\,{continues}\,{to}\,{\bar{L}}_{3},\,\alpha \lt \beta ,\end{array}\right.$ |
| • | ${\left[{G}_{2}\right]}_{2}\,{is}\,{analytic}\,{for}$ $\left\{\begin{array}{l}\eta \in {L}_{4},\,{and}\,{continues}\,{to}\,{\bar{L}}_{4},\,\alpha \gt \beta ,\\ \eta \in {L}_{1},\,{and}\,{continues}\,{to}\,{\bar{L}}_{1},\,\alpha \lt \beta ,\end{array}\right.$ |
| • | ${\left[{G}_{3}\right]}_{1}\,{is}\,{analytic}\,{for}$ $\left\{\begin{array}{l}\eta \in {L}_{3}\cup {L}_{4}\,{and}\,{continues}\,{to}\,{\bar{L}}_{3}\cup {\bar{L}}_{4},\alpha \gt \beta ,\\ \eta \in {L}_{1}\cup {L}_{2}\,{and}\,{continues}\,{to}\,{\bar{L}}_{1}\cup {\bar{L}}_{2},\alpha \lt \beta ,\end{array}\right.$ |
| • | ${\left[{G}_{3}\right]}_{2}\,{is}\,{analytic}\,{for}$ $\left\{\begin{array}{l}\eta \in {L}_{1}\cup {L}_{2}\,{and}\,{continues}\,{to}\,{\bar{L}}_{1}\cup {\bar{L}}_{2},\alpha \gt \beta ,\\ \eta \in {L}_{3}\cup {L}_{4}\,{and}\,{continues}\,{to}\,{\bar{L}}_{3}\cup {\bar{L}}_{4},\alpha \lt \beta ,\end{array}\right.$ |
| • | As η → ∞ , ${\left[{G}_{j}\right]}_{1}(x,t,\eta )\to {\left(\mathrm{1,0}\right)}^{{\rm{T}}}$, ${\left[{G}_{j}\right]}_{2}(x,t,\eta )\to {\left(\mathrm{0,1}\right)}^{{\rm{T}}}.$ |
Indeed, according to the definition of function Gj(x, t, η) in equation (
To better analyze special functions $\Psi$(η) and φ(η), one can get the following proposition according to the ISM theory.
It follows from equation (
$\begin{eqnarray}\psi (\eta )={\boldsymbol{I}}-{\int }_{0}^{\infty }{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}(\xi -x){\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,0,\eta ){\rm{d}}\xi ,\end{eqnarray}$
$\begin{eqnarray}{\phi }^{-1}(\eta )={\boldsymbol{I}}+{\int }_{0}^{T}{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(\tau -t){\hat{\sigma }}_{3}}({N}_{1}{G}_{1})(0,\tau ,\eta ){\rm{d}}\tau .\end{eqnarray}$
Assume that $\Psi$(η), φ(η) possess the following 2 × 2 matrix from, respectively2.28 ) and (2.33a ), (2.33b ) that the following key properties are ture
$\begin{eqnarray}\psi (\eta )=\left(\begin{array}{cc}\overline{f(\bar{\eta })} & s(\eta )\\ \overline{s(\bar{\eta })} & f(\eta )\end{array}\right),\phi (\eta )=\left(\begin{array}{cc}\overline{F(\bar{\eta })} & S(\eta )\\ \overline{S(\bar{\eta })} & F(\eta )\end{array}\right).\end{eqnarray}$
It follows from equations (| • | $\left(\begin{array}{c}s(\zeta )\\ f(\bar{\zeta })\end{array}\right)={\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(0,0,\eta )$ $=\,\left(\begin{array}{c}{\left({G}_{3}\right)}_{12}^{{L}_{1}\cup {L}_{2}}(0,t,\eta )\\ {\left({G}_{3}\right)}_{22}^{{L}_{1}\cup {L}_{2}}(0,t,\eta )\end{array}\right),$ |
| • | $\left(\begin{array}{c}-{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}T}S(\eta )\\ \overline{F(\bar{\eta })})\end{array}\right)={\left[{G}_{1}\right]}_{2}^{{L}_{2}\cup {L}_{4}}(t,\eta )$ $=\,\left(\begin{array}{c}{\left({G}_{1}\right)}_{12}^{{L}_{2}\cup {L}_{4}}(t,\eta )\\ {\left({G}_{1}\right)}_{22}^{{L}_{2}\cup {L}_{4}}(t,\eta )\end{array}\right),$ |
| • | f( − η) = f(η), s( − η) = − s(η), |
| • | F( − η) = F(η), S( − η) = − S(η), |
| • | $\det \psi (\eta )=f(\eta )\overline{f(\bar{\eta })}-s(\eta )\overline{s(\bar{\eta })}=1,\quad {for}\,\eta \in {\mathbb{R}},$ |
| • | $\det \phi (\eta )=1,{for}\,\eta \in {\mathbb{C}}\,(\mathrm{Im}\tfrac{2{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}=0,{if}\,T=\infty ),$ |
| • | f(η) = 1 + O(η−1), s(η) = O(η−1), ${as}\,\eta \to \infty ,\mathrm{Im}\tfrac{{\eta }^{2}}{\alpha -\beta }\gt 0,$ |
| • | F(η) = 1 + O(η−1), S(η) = O(η−1), ${as}\,\eta \to \infty ,\mathrm{Im}\tfrac{2{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}\gt 0.$ |
2.4. The basic RH problem
To facilitate subsequent calculations, we remember that the following symbolic assumptions
$\begin{eqnarray}\begin{array}{rcl}\mu (\eta ) & = & \frac{{\eta }^{2}}{\alpha -\beta }x+\frac{2{\eta }^{4}}{{\left(\alpha -\beta \right)}^{2}}t,\\ h(\eta ) & = & f(\eta )\overline{F(\bar{\eta })}-s(\eta )\overline{S(\bar{\eta })},\\ g(\eta ) & = & \overline{f(\bar{\eta })}\overline{S(\bar{\eta })}-\overline{s(\bar{\eta })}\overline{F(\bar{\eta })},\\ \theta (\eta ) & = & \frac{s(\eta )}{\overline{f(\bar{\eta })}},{\rm{\Theta }}(\eta )=-\frac{\overline{S(\bar{\eta })}}{f(\eta )h(\eta )},\end{array}\end{eqnarray}$
then, one obtains $\begin{eqnarray*}\begin{array}{rcl} & & \overline{S(\bar{\eta })}=f(\eta )g(\eta )+\overline{s(\bar{\eta })}h(\eta ),\\ & & h(\eta )\overline{h(\bar{\eta })}-g(\eta )\overline{g(\bar{\eta })}=1,\\ & & h(-\eta )=h(\eta ),g(-\eta )=-g(\eta ),\\ & & h(\eta )=1+O\left(\displaystyle \frac{1}{\eta }\right),g(\eta )=O\left(\displaystyle \frac{1}{\eta }\right){\rm{as}}\,\eta \to \infty ,\end{array}\end{eqnarray*}$
and the W(x, t, η) is defined by $\begin{eqnarray}\begin{array}{l}{\rm{for}}\,\alpha \gt \beta ,\\ \left\{\begin{array}{l}{W}_{+}(x,t,\eta )=\left(\tfrac{{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta )}{f(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\right),\eta \in {L}_{1},\\ {W}_{-}(x,t,\eta )=\left(\tfrac{{\left[{G}_{2}\right]}_{1}^{{L}_{2}}(x,t,\eta )}{h(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\right),\eta \in {L}_{2},\\ {W}_{-}(x,t,\eta )=\left({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\tfrac{{\left[{G}_{1}\right]}_{2}^{{L}_{3}}(x,t,\eta )}{\overline{f(\bar{\eta })}}\right),\eta \in {L}_{3},\\ {W}_{+}(x,t,\eta )=({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\tfrac{{\left[{G}_{2}\right]}_{2}^{{L}_{4}}(x,t,\eta )}{\overline{h(\bar{\eta })}}),\eta \in {L}_{4},\end{array}\right.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\rm{for}}\,\alpha \lt \beta ,\\ \left\{\begin{array}{l}{W}_{+}(x,t,\eta )=\left({\left[{G}_{3}\right]}_{1}^{{L}_{1}\cup {L}_{2}}(x,t,\eta ),\tfrac{{\left[{G}_{2}\right]}_{2}^{{L}_{1}}(x,t,\eta )}{\overline{h(\bar{\eta })}}\right),\eta \in {L}_{1},\\ {W}_{-}(x,t,\eta )=\left({\left[{G}_{3}\right]}_{1}^{{L}_{1}\cup {L}_{2}}(x,t,\eta ),\tfrac{{\left[{G}_{1}\right]}_{2}^{{L}_{2}}(x,t,\eta )}{\overline{f(\bar{\eta })}}\right),\eta \in {L}_{2},\\ {W}_{-}(x,t,\eta )=\left(\tfrac{{\left[{G}_{2}\right]}_{1}^{{L}_{3}}(x,t,\eta )}{h(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{3}\cup {L}_{4}}(x,t,\eta )\right),\eta \in {L}_{3},\\ {W}_{+}(x,t,\eta )=\left(\tfrac{{\left[{G}_{1}\right]}_{1}^{{L}_{4}}(x,t,\eta )}{f(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{3}\cup {L}_{4}}(x,t,\eta )\right),\eta \in {L}_{4}.\end{array}\right.\end{array}\end{eqnarray}$
These definitions imply that $\begin{eqnarray}\det W(x,t,\eta )=1,W(x,t,\eta )\to {\boldsymbol{I}},\eta \to \infty .\end{eqnarray}$
In the following, one only gives the case of α > β for jump condition and residue relation, and we can discuss the case of α < β similarly.
For α > β, set $q(x,t)\in {\mathbb{S}}$, and the function W(x, t, η) is given by equation (
$\begin{eqnarray}{W}_{-}(x,t,\eta )={W}_{+}(x,t,\eta )H(x,t,\eta ),\eta \in {\bar{L}}_{k},k=1,\ldots ,4,\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}H(x,t,\eta )\\ =\,\left\{\begin{array}{l}{H}_{1}(x,t,\eta ),\qquad \qquad \qquad \arg \tfrac{{\eta }^{2}}{\alpha -\beta }=\tfrac{\pi }{2},\\ {H}_{2}(x,t,\eta )={H}_{3}{H}_{4}^{-1}{H}_{1},\quad \arg \tfrac{{\eta }^{2}}{\alpha -\beta }=\pi ,\\ {H}_{3}(x,t,\eta ),\qquad \qquad \qquad \arg \tfrac{{\eta }^{2}}{\alpha -\beta }=\tfrac{3\pi }{2},\\ {H}_{4}(x,t,\eta ),\qquad \qquad \qquad \arg \tfrac{{\eta }^{2}}{\alpha -\beta }=0,\end{array}\right.\end{array}\end{eqnarray}$
and $\begin{eqnarray*}\begin{array}{rcl}{H}_{1}(x,t,\eta ) & = & \left(\begin{array}{cc}1 & 0\\ {\rm{\Theta }}(\eta ){{\rm{e}}}^{2{\rm{i}}\mu (\eta )} & 1\end{array}\right),\\ {H}_{3}(x,t,\eta ) & = & \left(\begin{array}{cc}1 & \overline{{\rm{\Theta }}(\bar{\eta })}{{\rm{e}}}^{-2{\rm{i}}\mu (\eta )}\\ 0 & 1\end{array}\right),\\ {H}_{4}(x,t,\eta ) & = & \left(\begin{array}{cc}1 & -\theta (\eta ){{\rm{e}}}^{-2{\rm{i}}\mu (\eta )}\\ \overline{\theta (\bar{\eta })}{{\rm{e}}}^{2{\rm{i}}\mu (\eta )} & 1-| \theta (\eta ){| }^{2}\end{array}\right).\end{array}\end{eqnarray*}$
From equations (
$\begin{eqnarray}\begin{array}{l}\overline{f(\bar{\eta })}{[{G}_{1}]}_{1}^{{L}_{1}}(x,t,\eta )-\overline{s(\bar{\eta })}{{\rm{e}}}^{2{\rm{i}}\mu (\eta )}{[{G}_{1}]}_{2}^{{L}_{3}}(x,t,\eta )\\ \quad =\,{[{G}_{3}]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}s(\eta ){{\rm{e}}}^{-2{\rm{i}}\mu (\eta )}{[{G}_{1}]}_{1}^{{L}_{1}}(x,t,\eta )+f(\eta ){[{G}_{1}]}_{2}^{{L}_{3}}(x,t,\eta )\\ \quad =\,{[{G}_{3}]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta ),\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}\overline{F(\bar{\eta })}{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta )-\overline{S(\bar{\eta })}{{\rm{e}}}^{2{\rm{i}}\mu (\eta )}{\left[{G}_{1}\right]}_{2}^{{L}_{3}}(x,t,\eta )\\ \quad =\,{\left[{G}_{2}\right]}_{1}^{{L}_{2}}(x,t,\eta ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}S(\eta ){{\rm{e}}}^{-2{\rm{i}}\mu (\eta )}{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta )+F(\eta ){\left[{G}_{1}\right]}_{2}^{{L}_{3}}(x,t,\eta )\\ \quad =\,{\left[{G}_{2}\right]}_{2}^{{L}_{4}}(x,t,\eta ),\end{array}\end{eqnarray}$
then, the equations ( $\begin{eqnarray}\begin{array}{l}h(\eta ){\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta )-g(\eta ){{\rm{e}}}^{2{\rm{i}}\mu (\eta )}{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\\ \quad =\,{\left[{G}_{2}\right]}_{1}^{{L}_{2}}(x,t,\eta ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\overline{g(\bar{\eta })}{{\rm{e}}}^{-2{\rm{i}}\mu (\eta )}{\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta )+\overline{h(\bar{\eta })}{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\\ \quad =\,{\left[{G}_{2}\right]}_{2}^{{L}_{4}}(x,t,\eta ).\end{array}\end{eqnarray}$
It follows from the equations ( $\begin{eqnarray}\begin{array}{l}\left(\displaystyle \frac{{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta )}{f(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\right)\\ \quad =\,\left(\displaystyle \frac{{\left[{G}_{2}\right]}_{1}^{{L}_{2}}(x,t,\eta )}{h(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\right){H}_{1}(x,t,\eta ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\displaystyle \frac{{\left[{G}_{2}\right]}_{2}^{{L}_{4}}(x,t,\eta )}{\overline{h(\bar{\eta })}}\right)\\ \quad =\,\left(\displaystyle \frac{{\left[{G}_{2}\right]}_{1}^{{L}_{2}}(x,t,\eta )}{h(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\right){H}_{2}(x,t,\eta ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\displaystyle \frac{{\left[{G}_{2}\right]}_{2}^{{L}_{4}}(x,t,\eta )}{\overline{h(\bar{\eta })}}\right)\\ \quad =\,\left({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\displaystyle \frac{{\left[{G}_{1}\right]}_{2}^{{L}_{3}}(x,t,\eta )}{\overline{f(\bar{\eta })}}\right){H}_{3}(x,t,\eta ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left(\displaystyle \frac{{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta )}{f(\eta )},{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,\eta )\right)\\ \quad =\,\left({\left[{G}_{3}\right]}_{1}^{{L}_{3}\cup {L}_{4}}(x,t,\eta ),\displaystyle \frac{{\left[{G}_{1}\right]}_{2}^{{L}_{3}}(x,t,\eta )}{\overline{f(\bar{\eta })}}\right){H}_{4}(x,t,\eta ).\end{array}\end{eqnarray}$
Therefore, the equations (One makes assumptions about the simple zeros of functions f(η) and h(η) as follows
| • | f(η) enjoys 2a simple zeros ${\{{\varsigma }_{j}\}}_{j=1}^{2a}$, 2a = 2a1 + 2a2. For α > β, if ${\{{\varsigma }_{j}\}}_{1}^{2{a}_{1}}\in {L}_{1}$, then ${\{{\bar{\varsigma }}_{j}\}}_{1}^{2{a}_{2}}\in {L}_{3}$. For α < β, if ${\{{\varsigma }_{j}\}}_{1}^{2{a}_{1}}\in {L}_{4}$, then ${\{{\bar{\varsigma }}_{j}\}}_{1}^{2{a}_{2}}\in {L}_{2}$. |
| • | h(η) enjoys 2b simple zeros ${\{{\zeta }_{j}\}}_{j=1}^{2b}$, 2b = 2b1 + 2b2. For α > β, if ${\{{\zeta }_{j}\}}_{1}^{2{b}_{1}}\in {L}_{4}$, then ${\{{\bar{\zeta }}_{j}\}}_{1}^{2{b}_{2}}\in {L}_{2}$. For α < β, if ${\{{\zeta }_{j}\}}_{1}^{2{b}_{1}}\in {L}_{1}$, then ${\{{\bar{\zeta }}_{j}\}}_{1}^{2{b}_{2}}\in {L}_{3}$. |
| • | The intersection of simple zeros of h(η) and f(η) is empty. |
Let $\dot{h}(\eta )=\tfrac{{\rm{d}}{h}}{{\rm{d}}\eta }$, one enjoys the following residue conditions:
$\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[W(x,t,\eta )\right]}_{1},{\varsigma }_{j}\}\\ =\,\displaystyle \frac{1}{s({\varsigma }_{j})\dot{f}({\varsigma }_{j})}{{\rm{e}}}^{2{\rm{i}}\mu ({\varsigma }_{j})}{\left[W(x,t,{\varsigma }_{j})\right]}_{2},j=1,\cdots ,\ 2{a}_{1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[W(x,t,\eta )\right]}_{2},{\bar{\varsigma }}_{j}\}\\ =\,-\displaystyle \frac{1}{\overline{s({\varsigma }_{j})}\overline{\dot{f}({\varsigma }_{j})}}{{\rm{e}}}^{-2{\rm{i}}\mu (\bar{{\varsigma }_{j}})}{\left[W(x,t,{\bar{\varsigma }}_{j})\right]}_{1},j=1,\cdots ,\ 2{a}_{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[W(x,t,\eta )\right]}_{1},{\zeta }_{j}\}\\ =\,-\displaystyle \frac{\overline{S({\bar{\zeta }}_{j})}}{f({\zeta }_{j})\dot{h}({\zeta }_{j})}{{\rm{e}}}^{2{\rm{i}}\mu ({\zeta }_{j})}{\left[W(x,t,{\zeta }_{j})\right]}_{1},j=1,\cdots ,\ 2{b}_{1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[W(x,t,\eta )\right]}_{2},{\bar{\zeta }}_{j}\}\\ =\,\displaystyle \frac{S({\bar{\zeta }}_{j})}{\overline{f({\zeta }_{j})}\overline{\dot{h}({\zeta }_{j})}}{{\rm{e}}}^{-2{\rm{i}}\mu ({\bar{\zeta }}_{j})}{\left[W(x,t,\bar{{\zeta }_{j}})\right]}_{2},j=1,\cdots ,\ 2{b}_{2}.\end{array}\end{eqnarray}$
One only shows the equation (
$\begin{eqnarray}\begin{array}{l}\mathrm{Res}\left\{\displaystyle \frac{{G}_{1}^{{L}_{1}}(x,t,\eta )}{f(\eta )},{\varsigma }_{j}\right\}\\ \quad =\,\mathop{\mathrm{lim}}\limits_{\eta \to {\varsigma }_{j}}(\eta -{\varsigma }_{j})\displaystyle \frac{{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta )}{f(\eta )}=\displaystyle \frac{{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,{\varsigma }_{j})}{\dot{f}({\varsigma }_{j})},\end{array}\end{eqnarray}$
taking η = ςj into the first and second equations of ( $\begin{eqnarray}{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,{\varsigma }_{j})=\displaystyle \frac{1}{s({\varsigma }_{j})}{{\rm{e}}}^{2{\rm{i}}\mu ({\varepsilon }_{j})}{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,{\varsigma }_{j}),\end{eqnarray}$
together with equations ( $\begin{eqnarray}\begin{array}{l}\mathrm{Res}\left\{\displaystyle \frac{{\left[{G}_{1}\right]}_{1}^{{L}_{1}}(x,t,\eta )}{f(\eta )},{\varsigma }_{j}\right\}\\ \quad =\ \displaystyle \frac{1}{s({\varsigma }_{j})\dot{f}({\varsigma }_{j})}{{\rm{e}}}^{2{\rm{i}}\mu ({\varsigma }_{j})}{\left[{G}_{3}\right]}_{2}^{{L}_{1}\cup {L}_{2}}(x,t,{\varsigma }_{j}),\end{array}\end{eqnarray}$
therefore, the equation (2.5. The inverse problem
The inverse problem includes the reconstruction of potential function q(x, t) from spectral functions ${\{{G}_{j}(x,t,\eta )\}}_{1}^{3}$. It follows from equation (2.10 ) that ${D}_{1}^{({\rm{od}})}=\tfrac{{\rm{i}}}{2}(\alpha -\beta ){{QD}}_{0}{\sigma }_{3}$. Since asymptotic expansion in equation (2.8 ) is a solution of equation (2.7 ), which implies that2.15 ) and given by2.16 ) if ${w}_{21}^{(1)}(x,t)$ replaces of w(x, t). It follows from equation (2.49 ) and its complex conjugate that2.13 ) can be expressed by w(x, t)
$\begin{eqnarray}q(x,t)=-\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }w(x,t){{\rm{e}}}^{-2{\rm{i}}{\int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}},\end{eqnarray}$
where G(x, t, η) is related to $\Psi$(x, t, η) as shown in equation ( $\begin{eqnarray*}\begin{array}{rcl}G(x,t,\eta ) & = & {\boldsymbol{I}}+\displaystyle \frac{{w}^{(1)}(x,t)}{\eta }+\displaystyle \frac{{w}^{(2)}(x,t)}{{\eta }^{2}}\\ & & +O\left(\displaystyle \frac{1}{{\eta }^{3}}\right),\eta \to \infty .\end{array}\end{eqnarray*}$
Meanwhile, G(x, t, η) is the solution of equation ( $\begin{eqnarray*}\begin{array}{rcl} & & q\overline{q}=\displaystyle \frac{4}{{\left(\alpha -\beta \right)}^{2}}| w{| }^{2},\\ & & \overline{q}{q}_{x}-q{\overline{q}}_{x}=\displaystyle \frac{4}{{\left(\alpha -\beta \right)}^{2}}(\overline{w}{w}_{x}-w{\overline{w}}_{x})+\displaystyle \frac{8\alpha }{{\left(\alpha -\beta \right)}^{2}}| w{| }^{4}.\end{array}\end{eqnarray*}$
Then, the one-form Ω given by equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{\Omega }} & = & \displaystyle \frac{\alpha }{{\left(\alpha -\beta \right)}^{2}}| w{| }^{2}{\rm{d}}x\\ & & -\left[\displaystyle \frac{\alpha }{{\left(\alpha -\beta \right)}^{2}}(\overline{w}{w}_{x}-w{\overline{w}}_{x})-\displaystyle \frac{6\alpha \beta -4{\beta }^{2}}{{\left(\alpha -\beta \right)}^{4}}| w{| }^{4}\right]{\rm{d}}t.\end{array}\end{eqnarray}$
Hence, one can solve the inverse problem according to the following steps successively:| i | (i) One utilizes any one of the functions ${\{{G}_{j}(x,t,\eta )\}}_{1}^{3}$ to calculate w(x, t) by $\begin{eqnarray*}w(x,t)=\mathop{\mathrm{lim}}\limits_{\lambda \to \infty }{\left[\eta {G}_{j}(x,t,\eta )\right]}_{21}.\end{eqnarray*}$ |
| ii | (ii) One gets Ω(x, t) from equation ( |
| iii | (iii) One computes potential function q(x, t) by equation ( |
2.6. The global relation
In this subsection, one gives the spectral functions f(η), s(η), F(η), S(η) which are not independent but admit a significant relationship. In fact, at the boundary of the region (ξ, τ): 0 < ξ < ∞, 0 < τ < t, the integral of the one-form A(x, t, η) defined by the equation (2.17 ) is vanished. Let G(x, t, η) = G3(x, t, η) in the one-form A(x, t, η), one obtains2.31b ), one can find that the first term of the equation (2.51 ) is2.27a ), we obtain
$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{\infty }^{0}{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}\xi {\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,0,\eta ){\rm{d}}\xi \\ \quad +\,{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\tau {\hat{\sigma }}_{3}}({N}_{1}{G}_{3})(0,\tau ,\eta ){\rm{d}}\tau \\ \quad +\,{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t{\hat{\sigma }}_{3}}{\displaystyle \int }_{0}^{\infty }{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}\xi {\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,t,\eta ){\rm{d}}\xi \\ \ =\,\mathop{\mathrm{lim}}\limits_{x\to \infty }{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}x{\hat{\sigma }}_{3}}{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\tau {\hat{\sigma }}_{3}}({N}_{1}{G}_{3})(x,\tau ,\eta ){\rm{d}}\tau .\end{array}\end{eqnarray}$
On the one hand, since $\Psi$(η) = G3(0, 0, η), together with equation ( $\begin{eqnarray*}\psi (\eta )-{\boldsymbol{I}}.\end{eqnarray*}$
Set x = 0 in the equation ( $\begin{eqnarray}{G}_{3}(0,\tau ,\eta )={G}_{1}(0,\tau ,\eta ){{\rm{e}}}^{-\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\tau {\hat{\sigma }}_{3}}\psi (\eta ),\end{eqnarray}$
then $\begin{eqnarray}\begin{array}{l}{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\tau {\hat{\sigma }}_{3}}({N}_{1}{G}_{3})(0,\tau ,\eta )\\ \quad =\,\left[{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\tau {\hat{\sigma }}_{3}}({M}_{1}{G}_{1})(0,\tau ,\eta )\right]\psi (\eta ).\end{array}\end{eqnarray}$
On the other hand, it follows from equations (2.53 ) and (2.30a ) that the second term of the equation (2.51 ) is
$\begin{eqnarray*}\begin{array}{l}{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\tau {\hat{\sigma }}_{3}}({N}_{1}{G}_{3})(0,\tau ,\eta ){\rm{d}}\tau \\ \quad =\,\left[{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t{\hat{\sigma }}_{3}}{N}_{1}{G}_{1}(0,t,\eta )-I\right]\psi (\eta ).\end{array}\end{eqnarray*}$
Let $q(x,t)\in {\mathbb{S}}$ for x → ∞ , then, equation (2.51 ) turns into2.54 ) is valid for η2 in the lower half-plane and the second column of equation (2.54 ) is valid for η2 in the upper half-plane, and the expression of φ(t, η) is2.54 ) turns into2.55 ) is2.56 ) is the so-called global relation.
$\begin{eqnarray}\begin{array}{l}{\phi }^{-1}(t,\eta )\psi (\eta )+{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t{\hat{\sigma }}_{3}}\\ \quad \times \,{\displaystyle \int }_{0}^{\infty }{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}\xi {\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,t,\eta ){\rm{d}}\xi ={\boldsymbol{I}},\end{array}\end{eqnarray}$
where the first column of equation ( $\begin{eqnarray*}{\phi }^{-1}(t,\eta )={{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t{\hat{\sigma }}_{3}}{G}_{1}(0,t,\eta ).\end{eqnarray*}$
Denoting φ(η) = φ(T, η) and letting t = T, one finds that the equation ( $\begin{eqnarray}\begin{array}{l}{\phi }^{-1}(\eta )\psi (\eta )+{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}T{\hat{\sigma }}_{3}}\\ \quad \times \,{\displaystyle \int }_{0}^{\infty }{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}\xi {\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,T,\eta ){\rm{d}}\xi ={\boldsymbol{I}}.\end{array}\end{eqnarray}$
Hence, the (21)-component of equation ( $\begin{eqnarray}f(\eta )S(\eta )-F(\eta )s(\eta )={{\rm{e}}}^{\tfrac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}T}E(\eta ),\mathrm{Im}{\eta }^{2}\geqslant 0,\end{eqnarray}$
where E(η) is expressed by $\begin{eqnarray}E(\eta )={\int }_{0}^{\infty }{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}\xi }{\left({M}_{1}{G}_{3}\right)}_{21}(\xi ,T,\eta ){\rm{d}}\xi .\end{eqnarray}$
Indeed, equation (3. The functions f(η), s(η), F(η) and S(η)
(f(η) and s(η)) Let ${u}_{0}(x)=u(x,0)\in {\mathbb{S}}$, one defines the mapping
$\begin{eqnarray*}{{\mathbb{Y}}}_{1}\,:\{{u}_{0}(x)\}\to \{f(\eta ),s(\eta )\},\end{eqnarray*}$
in terms of $\begin{eqnarray*}{\left(s(\eta ),f(\eta )\right)}^{{\rm{T}}}=\left\{\begin{array}{l}{[{G}_{3}]}_{2}^{{L}_{1}\cup {L}_{2}}(x,0,\eta ),\,{\rm{for}}\,\alpha \gt \beta ,\\ {[{G}_{3}]}_{2}^{{L}_{3}\cup {L}_{4}}(x,0,\eta ),\,{\rm{for}}\,\alpha \lt \beta ,\end{array}\right.\end{eqnarray*}$
where G3(x, 0, η) is given by $\begin{eqnarray*}{G}_{3}(x,0,\eta )={\boldsymbol{I}}-{\int }_{x}^{\infty }{{\rm{e}}}^{\tfrac{{\rm{i}}}{\alpha -\beta }{\eta }^{2}(\xi -x){\hat{\sigma }}_{3}}({M}_{1}{G}_{3})(\xi ,0,\eta ){\rm{d}}\xi ,\end{eqnarray*}$
with M1(x, 0, η) expressed by equation (The f(η) and s(η) possess the properties as following
| i | (i) f(η), s(η) are analytic and bounded for $\mathrm{Im}\tfrac{1}{\alpha -\beta }{\eta }^{2}\gt 0$ and continuous for $\mathrm{Im}\tfrac{1}{\alpha -\beta }{\eta }^{2}\geqslant 0$. |
| ii | (ii) $f(\eta )=1\,+\,O\left(\tfrac{1}{\eta }\right),s(\eta )=O\left(\tfrac{1}{\eta }\right)$ as η → ∞, $\mathrm{Im}\tfrac{1}{\alpha -\beta }{\eta }^{2}\geqslant 0$. |
| iii | (iii) $f(\eta )\overline{f(\bar{\eta })}-s(\eta )\overline{s(\bar{\eta })}=1$, ${\eta }^{2}\in {\mathbb{R}}$. |
| iv | (iv) f(− η) = f(η), s(− η) = − s(η), $\mathrm{Im}\tfrac{1}{\alpha -\beta }{\eta }^{2}\geqslant 0$. |
| v | (v) The inverse mapping of ${{\mathbb{Y}}}_{1}$ is ${{\mathbb{Y}}}_{1}^{-1}={{\mathbb{Z}}}_{1}\,:\{f(\eta ),s(\eta )\}\,\to \{{u}_{0}(x)\}$, which is defined by $\begin{eqnarray*}\begin{array}{rcl}{u}_{0}(x) & = & -\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }w(x){{\rm{e}}}^{-2{\rm{i}}\alpha {\displaystyle \int }_{0}^{x}| w(\xi ){| }^{2}{\rm{d}}\xi },\\ w(x) & = & \mathop{\mathrm{lim}}\limits_{\eta \to \infty }{\left[\eta {W}^{(x)}(x,\eta )\right]}_{21},\end{array}\end{eqnarray*}$ where W(x)(x, η) admits RH problem as follows. |
| • | ${W}^{(x)}(x,\eta )=\left\{\begin{array}{l}{W}_{-}^{(x)}(x,\eta ),\mathrm{Im}\tfrac{1}{\alpha -\beta }{\eta }^{2}\leqslant 0,\\ {W}_{+}^{(x)}(x,\eta ),\mathrm{Im}\tfrac{1}{\alpha -\beta }{\eta }^{2}\geqslant 0,\end{array}\right.$ is a section analytic function. |
| • | ${W}_{-}^{(x)}(x,\eta )={W}_{+}^{(x)}(x,\eta ){\left({H}^{(x)}(x,\eta )\right)}^{-1}$, ${\eta }^{2}\in {\mathbb{R}}$, and $\begin{eqnarray}{H}^{(x)}(x,\eta )=\left(\begin{array}{cc}1 & -\theta (\eta ){{\rm{e}}}^{-\tfrac{2{\rm{i}}}{\alpha -\beta }{\eta }^{2}x}\\ \overline{\theta (\bar{\eta })}{{\rm{e}}}^{\tfrac{2{\rm{i}}}{\alpha -\beta }{\eta }^{2}x} & 1-| \theta (\eta ){| }^{2}\end{array}\right).\end{eqnarray}$ |
| • | ${W}^{(x)}(x,\eta )={\boldsymbol{I}}+O\left(\tfrac{1}{\eta }\right),\eta \to \infty .$ |
| • | f(η) possesses 2a simple zeros ${\{{\varsigma }_{j}\}}_{1}^{2a}$, 2a = 2a1 + 2a2, such that $\mathrm{Im}\tfrac{1}{\alpha -\beta }{\varsigma }_{j}^{2}\gt 0,j=1,2,\ \cdots ,\ 2{a}_{1}$, and $\mathrm{Im}\tfrac{1}{\alpha -\beta }{\varsigma }_{j}^{2}\,\lt 0,j=1,2,\ \cdots ,\ 2{a}_{2}$. |
| • | The first column of ${W}_{+}^{(x)}(x,\eta )$ enjoys simple poles at $\eta ={\{{\bar{\varsigma }}_{j}\}}_{1}^{2{a}_{2}}$. The second column of ${W}_{-}^{(x)}(x,\eta )$ enjoys simple poles at $\eta ={\{{\varsigma }_{j}\}}_{1}^{2{a}_{1}}$. The relevant residue expression is $\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[{W}^{(x)}(x,\eta )\right]}_{1},{\varsigma }_{j}\}\\ \quad =\,\displaystyle \frac{{{\rm{e}}}^{\tfrac{2{\rm{i}}}{\alpha -\beta }{\varsigma }_{j}^{2}x}}{\dot{f}({\varsigma }_{j})s({\varsigma }_{j})}{\left[{W}^{(x)}(x,{\varsigma }_{j})\right]}_{2},j=1,2,\ \cdots ,\ 2{a}_{1},\end{array}\end{eqnarray}$ $\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[{W}^{(x)}(x,\eta )\right]}_{2},{\bar{\varsigma }}_{j}\}\\ \quad =\displaystyle \frac{{{\rm{e}}}^{-\tfrac{2{\rm{i}}}{\alpha -\beta }{\overline{\varsigma }}_{j}^{2}x}}{\overline{\dot{f}({\varsigma }_{j})}\overline{s({\varsigma }_{j})}}{\left[{W}^{(x)}(x,{\bar{\varsigma }}_{j})\right]}_{1},j=1,2,\ \cdots ,\ 2{a}_{2}.\end{array}\end{eqnarray}$ |
(i)–(iv) follow from the investigation in section
(F(η) and S(η)) Let ${v}_{0}(t),{v}_{1}(t)\in {\mathbb{S}}$, the mapping
$\begin{eqnarray*}{{\mathbb{Y}}}_{2}\,:\{{v}_{0}(t),{v}_{1}(t)\}\to \{F(\eta ),S(\eta )\},\end{eqnarray*}$
in terms of $\begin{eqnarray*}{\left(S(\eta ),F(\eta )\right)}^{{\rm{T}}}=\left\{\begin{array}{l}{[{G}_{1}]}_{2}^{{L}_{1}}(x,0,\eta ),\,{\rm{for}}\,\alpha \gt \beta ,\\ {[{G}_{1}]}_{2}^{{L}_{4}}(x,0,\eta ),\,{\rm{for}}\,\alpha \lt \beta ,\end{array}\right.\end{eqnarray*}$
where G1(0, t, η) is given by $\begin{eqnarray*}{G}_{1}(0,t,\eta )={\boldsymbol{I}}-{\int }_{t}^{T}{{\rm{e}}}^{\tfrac{2{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}(\tau -t){\hat{\sigma }}_{3}}({N}_{1}{G}_{1})(0,\tau ,\eta ){\rm{d}}\tau ,\end{eqnarray*}$
and N1(0, t, η) is expressed by equation (The F(η) and S(η) possess the properties as follows
| i | (i) F(η), S(η) are analytic and bounded for $\mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\,\geqslant 0$, if T = ∞, the F(η), S(η) are defined only for $\mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\geqslant 0$. |
| ii | (ii) $F(\eta )=1\,+\,O\left(\tfrac{1}{\eta }\right),S(\eta )=O\left(\tfrac{1}{\eta }\right)$ as η → ∞ , $\mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\geqslant 0$. |
| iii | (iii) $F(\eta )\overline{F(\bar{\eta })}-S(\eta )\overline{S(\bar{\eta })}=1$, $\eta \in {\mathbb{C}}({\eta }^{4}\in {\mathbb{R}},\,{if}\,T=\infty )$. |
| iv | (iv) F(−η) = F(η), S(−η) = − S(η), $\mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\geqslant 0$. |
| v | (v) The inverse mapping of ${{\mathbb{Y}}}_{2}$ is ${{\mathbb{Y}}}_{2}^{-1}={{\mathbb{Z}}}_{2}\,:\{F(\eta ),S(\eta )\}\to \{{v}_{0}(t),{v}_{1}(t)\}$, which is defined by $\begin{eqnarray}\begin{array}{rcl}{v}_{0}(t) & = & -\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }{w}_{12}^{(1)}(t){{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{0}^{t}{{\rm{\Omega }}}_{2}(\tau ){\rm{d}}\tau },\\ {v}_{1}(t) & = & \left[\displaystyle \frac{4}{{\left(\alpha -\beta \right)}^{2}}{w}_{21}^{(3)}(t)-{v}_{0}(t){\bar{v}}_{0}(t){w}_{21}^{(1)}(t)\right]{{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{0}^{t}{{\rm{\Omega }}}_{2}(\tau ){\rm{d}}\tau }\\ & & -\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }{v}_{0}(t){w}_{11}^{(2)}(t)-\displaystyle \frac{{\rm{i}}\alpha }{2}| {v}_{0}(t){| }^{2}{\bar{v}}_{0},\end{array}\end{eqnarray}$ where $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Omega }}}_{2}(\tau ) & = & 2(-{\alpha }^{2}+\alpha \beta -{\beta }^{2})| {w}_{21}^{(1)}{| }^{4}\\ & & +\displaystyle \frac{2\alpha }{{\left(\alpha -\beta \right)}^{3}}({\bar{w}}_{21}^{(1)}{w}_{21}^{(3)}+{w}_{21}^{(1)}{\bar{w}}_{21}^{(3)})\\ & & -\displaystyle \frac{4\alpha }{\alpha -\beta }| {w}_{21}^{(1)}{| }^{4}-\displaystyle \frac{4\alpha }{\alpha -\beta }| {w}_{21}^{(1)}{| }^{2}\mathrm{Re}[{w}_{11}^{(2)}],\end{array}\end{eqnarray*}$ and the functions w(j)(t), j = 1, 2, 3 are determined by $\begin{eqnarray*}\begin{array}{rcl}{W}^{(t)}(t,\eta ) & = & {\boldsymbol{I}}+\displaystyle \frac{{w}^{(1)}(t)}{\eta }+\displaystyle \frac{{w}^{(2)}(t)}{{\eta }^{2}}+\displaystyle \frac{{w}^{(3)}(t)}{{\eta }^{3}}\\ & & +O\left(\displaystyle \frac{1}{{\eta }^{4}}\right),\eta \to \infty ,\end{array}\end{eqnarray*}$ where W(t)(t, η) admits RH problem as follows |
| • | ${W}^{(t)}(t,\eta )=\left\{\begin{array}{ll}{W}_{-}^{(t)}(t,\eta ), & \mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\leqslant 0,\\ {W}_{+}^{(t)}(t,\eta ), & \mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}\geqslant 0,\end{array}\right.$ is a section analytic function. |
| • | ${W}_{-}^{(t)}(t,\eta )={W}_{+}^{(t)}(t,\eta ){H}^{(t)}(t,\eta )$, ${\eta }^{4}\in {\mathbb{R}}$, and $\begin{eqnarray}{H}^{(t)}(t,\eta )=\left(\begin{array}{cc}1 & -\tfrac{S(\eta )}{\overline{F(\overline{\eta })}}{{\rm{e}}}^{-\tfrac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t}\\ \tfrac{\overline{S(\overline{\eta })}}{F(\eta )}{{\rm{e}}}^{\tfrac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\eta }^{4}t} & \tfrac{1}{F(\eta )\overline{F(\overline{\eta })}}\end{array}\right).\end{eqnarray}$ |
| • | ${W}^{(t)}(T,\eta )={\boldsymbol{I}}+O\left(\tfrac{1}{\eta }\right),\eta \to \infty .$ |
| • | F(η) possesses 2k simple zeros ${\{{\varepsilon }_{j}\}}_{1}^{2k}$, 2k = 2k1 + 2k2 such that $\mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\varepsilon }_{j}^{4}\gt 0,j\,=\,1,2,\cdots ,2{k}_{1}$, and $\mathrm{Im}\tfrac{2}{{\left(\alpha -\beta \right)}^{2}}{\varepsilon }_{j}^{4}\lt 0,j\,=\,1,2,\cdots ,2{k}_{2}$. |
| • | The first column of ${W}_{+}^{(t)}(t,\eta )$ enjoys simple poles at $\eta ={\{{\bar{\varepsilon }}_{j}\}}_{1}^{2{k}_{2}}$, the second column of ${W}_{-}^{(t)}(t,\eta )$ enjoys simple poles at $\eta ={\{{\varepsilon }_{j}\}}_{1}^{2{k}_{2}}$. The relevant residue expression is $\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[{W}^{(t)}(t,\eta )\right]}_{1},{\varepsilon }_{j}\}\\ =\,\displaystyle \frac{{{\rm{e}}}^{\tfrac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\varepsilon }_{j}^{4}t}}{\dot{F}({\varepsilon }_{j})S({\varepsilon }_{j})}{\left[{W}^{(t)}(t,{\varepsilon }_{j})\right]}_{2},j=1,2,\cdots ,2{k}_{1},\end{array}\end{eqnarray}$ $\begin{eqnarray}\begin{array}{l}\mathrm{Res}\{{\left[{W}^{(t)}(t,\eta )\right]}_{2},{\bar{\varepsilon }}_{j}\}\\ =\,\displaystyle \frac{{{\rm{e}}}^{-\tfrac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{\overline{\varepsilon }}_{j}^{4}t}}{\overline{\dot{F}({\bar{\varepsilon }}_{j})}\overline{S({\bar{\varepsilon }}_{j})}}{\left[{W}^{(t)}(t,{\bar{\varepsilon }}_{j})\right]}_{1},j=1,2,\ \cdots ,\ 2{k}_{2}.\end{array}\end{eqnarray}$ |
(i)–(iv) follow from the investigate in section
4. The RH problem
Let ${u}_{0}(x)\in {\mathbb{S}}({{\rm{R}}}^{+})$, the matrix functions $\Psi$(η) and φ(η) in terms of f(η), s(η), F(η), S(η) are given by equation (
| • | W(x, t, η) is the slice analytic function for η ∈ Lk and continuous to ${\bar{L}}_{k},(k=1,\ \ldots ,\ 4)$. |
| • | W(x, t, η) jump arises on the curves ${\{{\bar{L}}_{k}\}}_{1}^{4}$ and admits the jump relation given by theorem $\begin{eqnarray*}{W}_{-}(x,t,\eta )={W}_{+}(x,t,\eta )H(x,t,\eta ),\eta \in {\bar{L}}_{k},k=1,\ \ldots ,\ 4,\end{eqnarray*}$ |
| • | $W(x,t,\eta )={\boldsymbol{I}}+{\rm{O}}\left(\tfrac{1}{\eta }\right),\eta \to \infty $. |
| • | W(x, t, η) meets the residue conditions given by proposition |
$\begin{eqnarray}\begin{array}{rcl} & & q(x,t)=-\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }w(x,t){{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}},\\ & & w(x,t)=\mathop{\mathrm{lim}}\limits_{\eta \to \infty }{\left[\eta W(x,t,\eta )\right]}_{21},\\ & & {\rm{\Omega }}=\displaystyle \frac{\alpha }{{\left(\alpha -\beta \right)}^{2}}| w{| }^{2}{\rm{d}}x\\ & & -\left[\displaystyle \frac{\alpha }{{\left(\alpha -\beta \right)}^{2}}(\overline{w}{w}_{x}-w{\overline{w}}_{x})-\displaystyle \frac{6\alpha \beta -4{\beta }^{2}}{{\left(\alpha -\beta \right)}^{4}}| w{| }^{4}\right]{\rm{d}}t,\end{array}\end{eqnarray}$
thus, the function q(x, t) is a solution of the GDNLS equation (Indeed, one can manifest the above RH problem following [4].
5. Conclusions and discussions
In this paper, we use UTM to discuss the IBVPs of the generalized DNLS equation (1.3 ), one can also discuss the equation (1.3 ) on a finite interval, and analyze the asymptotic behavior of the solution for the equation (1.3 ) by the Deift–Zhou method [36]. Since the RH problem is equivalent to Gel’fand–Levitan–Marchenko (GLM) theory, one can obtain the soliton solution of the equation (1.3 ) by solving the GLM equation following [37], which are our future investigation work.
Acknowledgments
This work is supported by the Natural Science Foundation of China (Nos. 11 601 055, 11 805 114 and 11 975 145), the Natural Science Research Projects of Anhui Province (No. KJ2019A0637), and University Excellent Talent Fund of Anhui Province (No. gxyq2019096).
Appendix. Recovering v0(t) and v1(t)
In this appendix, we will give a proof of equation (3.3 ), that is, derive v0(t) and v1(t) from W(t). Let G(x, t, η) is a solution of equation (2.16 ). According to equation (2.11 ), one gets2.7 ) and enjoys the following form2.15 ) and related to G(x, t, η) as followsA.1 ) gives2.49 ), one findsA.2 )–(A.4 ) that2.14 ) can be expressed asA.2 ), (A.4 )–(A.7 ) at x = 0 and yield
$\begin{eqnarray}\begin{array}{rcl}{\rm{i}}{Q}_{x}{\sigma }_{3}{D}_{0} & = & \displaystyle \frac{4{\rm{i}}}{{\left(\alpha -\beta \right)}^{2}}{D}_{4}^{({\rm{od}})}{\sigma }_{3}-{\rm{i}}{Q}^{2}{\sigma }_{3}{D}_{2}^{({\rm{od}})}\\ & & +\displaystyle \frac{2}{\alpha -\beta }{{QD}}_{3}^{(d)}+\displaystyle \frac{\alpha }{2}{Q}^{3}{D}_{0},\end{array}\end{eqnarray}$
where $\Psi$(x, t, η) is the solution of equation ( $\begin{eqnarray*}{\rm{\Psi }}(x,t,\eta )={D}_{0}+\displaystyle \frac{{D}_{1}}{\eta }+\displaystyle \frac{{D}_{2}}{{\eta }^{2}}+\displaystyle \frac{{D}_{3}}{{\eta }^{3}}+O\left(\displaystyle \frac{1}{{\eta }^{4}}\right),\eta \to \infty .\end{eqnarray*}$
Since $\Psi$(x, t, η) is defined by equation ( $\begin{eqnarray*}G(x,t,\eta )=\left(\begin{array}{cc}{G}_{11} & {G}_{12}\\ {G}_{21} & {G}_{22}\end{array}\right),\end{eqnarray*}$
then, one gets $\begin{eqnarray*}{\rm{\Psi }}(x,t,\eta )=\left(\begin{array}{cc}{D}_{0}^{11}{G}_{11} & {D}_{0}^{22}{{\rm{e}}}^{2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}}{G}_{12}\\ {D}_{0}^{11}{{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}}{G}_{21} & {D}_{0}^{22}{G}_{22}\end{array}\right).\end{eqnarray*}$
If seeking $\begin{eqnarray*}G(x,t,\eta )={\boldsymbol{I}}+\displaystyle \frac{{w}^{(1)}}{\eta }+\displaystyle \frac{{w}^{(2)}}{{\eta }^{2}}+\displaystyle \frac{{w}^{(3)}}{{\eta }^{3}}+O\left(\displaystyle \frac{1}{{\eta }^{4}}\right),\eta \to \infty ,\end{eqnarray*}$
then the (21)-entry of equation ( $\begin{eqnarray}\begin{array}{rcl}{q}_{x} & = & \left[\displaystyle \frac{4}{{\left(\alpha -\beta \right)}^{2}}{w}_{21}^{(3)}-q\bar{q}{w}_{21}^{(1)}\right]{{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}}\\ & & -\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }{{qw}}_{11}^{(2)}-\displaystyle \frac{{\rm{i}}\alpha }{2}{q}^{2}\bar{q}.\end{array}\end{eqnarray}$
Taking the complex conjugate yields $\begin{eqnarray}\begin{array}{rcl}{\bar{q}}_{x} & = & \left[\displaystyle \frac{4}{{\left(\alpha -\beta \right)}^{2}}{\bar{w}}_{21}^{(3)}-q\bar{q}{\bar{w}}_{21}^{(1)}\right]{{\rm{e}}}^{2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}}\\ & & +\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }\bar{q}{\bar{w}}_{11}^{(2)}+\displaystyle \frac{{\rm{i}}\alpha }{2}q{\bar{q}}^{2}.\end{array}\end{eqnarray}$
At the same time, from equation ( $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & -\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }{w}_{21}^{(1)}{{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}},\\ \bar{q}(x,t) & = & \displaystyle \frac{2{\rm{i}}}{\alpha -\beta }{\bar{w}}_{21}^{(1)}{{\rm{e}}}^{2{\rm{i}}{\displaystyle \int }_{(\mathrm{0,0})}^{(x,t)}{\rm{\Omega }}}.\end{array}\end{eqnarray}$
It follows from equations ( $\begin{eqnarray}\begin{array}{l}\bar{q}{q}_{x}-q{\bar{q}}_{x}=\displaystyle \frac{8{\rm{i}}}{{\left(\alpha -\beta \right)}^{3}}({\bar{w}}_{21}^{(1)}{w}_{21}^{(3)}+{w}_{21}^{(1)}{\bar{w}}_{21}^{(3)})\\ -\,\displaystyle \frac{4{\rm{i}}}{\alpha -\beta }q\bar{q}{w}_{21}^{(1)}{\bar{w}}_{21}^{(1)}-\displaystyle \frac{4{\rm{i}}}{\alpha -\beta }q\bar{q}\mathrm{Re}[{w}_{11}^{(2)}]-{\rm{i}}\alpha {q}^{2}{\bar{q}}^{2},\end{array}\end{eqnarray}$
which means that the coefficient ${{\rm{\Omega }}}_{2}=\tfrac{1}{8}({\alpha }^{2}+\alpha \beta \,-{\beta }^{2})| q{| }^{4}-\tfrac{{\rm{i}}}{4}\alpha (\bar{q}{q}_{x}-q{\bar{q}}_{x})$ of dt in the differential form Ω defined in equation ( $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}_{2} & = & \displaystyle \frac{1}{8}(-{\alpha }^{2}+\alpha \beta -{\beta }^{2})| q{| }^{4}\\ & & +\displaystyle \frac{2\alpha }{{\left(\alpha -\beta \right)}^{3}}({\bar{w}}_{21}^{(1)}{w}_{21}^{(3)}+{w}_{21}^{(1)}{\bar{w}}_{21}^{(3)})\\ & & -\displaystyle \frac{\alpha }{\alpha -\beta }q\bar{q}{w}_{21}^{(1)}{\bar{w}}_{21}^{(1)}-\displaystyle \frac{\alpha }{\alpha -\beta }q\bar{q}\mathrm{Re}[{w}_{11}^{(2)}].\end{array}\end{eqnarray}$
Owing to $q\bar{q}=4| {w}_{21}^{(1)}{| }^{2},$ we calculate equations ( $\begin{eqnarray}\begin{array}{rcl}{v}_{0}(t) & = & -\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }{w}_{12}^{(1)}(t){{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{0}^{t}{{\rm{\Omega }}}_{2}(\tau ){\rm{d}}\tau },\\ {v}_{1}(t) & = & \left[\displaystyle \frac{4}{{\left(\alpha -\beta \right)}^{2}}{w}_{21}^{(3)}(t)-{v}_{0}(t){\bar{v}}_{0}(t){w}_{21}^{(1)}(t)\right]{{\rm{e}}}^{-2{\rm{i}}{\displaystyle \int }_{0}^{t}{{\rm{\Omega }}}_{2}(\tau ){\rm{d}}\tau }\\ & & -\displaystyle \frac{2{\rm{i}}}{\alpha -\beta }{v}_{0}(t){w}_{11}^{(2)}(t)-\displaystyle \frac{{\rm{i}}\alpha }{2}| {v}_{0}(t){| }^{2}{\bar{v}}_{0},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}_{2}(\tau ) & = & 2(-{\alpha }^{2}+\alpha \beta -{\beta }^{2})| {w}_{21}^{(1)}{| }^{4}\\ & & +\displaystyle \frac{2\alpha }{{\left(\alpha -\beta \right)}^{3}}({\bar{w}}_{21}^{(1)}{w}_{21}^{(3)}+{w}_{21}^{(1)}{\bar{w}}_{21}^{(3)})\\ & & -\displaystyle \frac{4\alpha }{\alpha -\beta }| {w}_{21}^{(1)}{| }^{4}-\displaystyle \frac{4\alpha }{\alpha -\beta }| {w}_{21}^{(1)}{| }^{2}\mathrm{Re}[{w}_{11}^{(2)}],\end{array}\end{eqnarray}$
where the functions w(j)(t), j = 1, 2, 3 are determined by $\begin{eqnarray}\begin{array}{rcl}{W}^{(t)}(t,\eta ) & = & {\boldsymbol{I}}+\displaystyle \frac{{w}^{(1)}(t)}{\eta }+\displaystyle \frac{{w}^{(2)}(t)}{{\eta }^{2}}+\displaystyle \frac{{w}^{(3)}(t)}{{\eta }^{3}}\\ & & +O\left(\displaystyle \frac{1}{{\eta }^{4}}\right),\eta \to \infty .\end{array}\end{eqnarray}$