1. Introduction
Along with the study of rogue wave that ‘appear from nowhere and disappear without a trace' [1], the localized wave has drawn much attention in recent decades particularly from the point of generative mechanism [2]. As prototypical localized waves, the rogue wave solutions, as the parameter limits of breather solutions, of the focusing nonlinear Schrödinger equation had been presented in the rational form in 1983, which was also called the Peregrine soliton [3]. This soliton is of fundamental significance because it is localized in both time and space, and because it defines the limit of a wide class of solutions to the nonlinear Schrödinger equation. The mechanism via ‘resonance' procedure to generate rogue waves, usually appear as algebraic rational solutions, has been widely investigated by many researchers [4-6] (also see a recent review [7] and the references therein). Up to now, many methods have been used to search for localized solutions of the integrable systems, such as Darboux transformation, bilinear method, multilinear variable separation approach, physics-informed neural network method, and many others [8-13].
The localized wave, apart from rational solutions, can be also generated by introducing nonisospectral effects which are usually related to solitary waves in non-uniformity of media [14]. For the nonisospectral integrable equations, they are usually related to time-dependent spectral parameters, which determine the time-varying amplitude and velocity of the solitary waves. A series of works have demonstrated localized structure of solitons for nonisospectral soliton equations (see [14-19]). In [20], Silem et al discussed space-time localized waves of three nonisospectral soliton equations, including the nonisospectral Korteweg-de Vries, modified Korteweg-de Vries and Hirota equations. The idea is introducing a time-dependent spectral parameter λ with λt=α(t)λ, where α is a function of t, it is possible to get a decaying amplitude. Although these nonisospectral equations can be gauge-transformed to their isospectral counterparts, the mechanism to generate space-time localized solitary waves by introducing nonisospectral effects is interesting and worthy considered.
The sine-Gordon (sG) model arises in differential geometry and relativistic field theory. It also appears in a number of other physical applications, including the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendula attached to a stretched wire, and dislocations in crystals, and so on. In the present paper we shall focus on the following complex nonisospectral nonpotential sG (cnnsG) equation
$\begin{eqnarray}\begin{array}{rcl}{u}_{xt}+2u{\partial }_{x}^{-1}{(u{u}^{* })}_{t} & =& \alpha (t)[x({u}_{xx}+2{u}^{2}{u}^{* })\\ & & +2{u}_{x}+2u{\partial }_{x}^{-1}(u{u}^{* })]+u,\end{array}\end{eqnarray}$
and construct some localized solutions, where ${\partial }_{x}^{-1}\,=\frac{1}{2}\left({\int }_{-\infty }^{x}-{\int }_{x}^{+\infty }\right)\cdot \,\rm{d}\,x$ and asterisk denotes the complex conjugate. For the nonpotential sG equation, one can refer to [21]. The approach adopted in this paper is the so-called bilinearization reduction method, which is developed recently to investigate solutions for the nonlocal integrable systems [21-23].The paper is organized as follows. In section 2 , we introduce a nonisospectral negative order Ablowitz-Kaup-Newell-Segur (nnAKNS) equation and discuss its Lax integrability. Moreover, double Wronskian solution to the nnAKNS equation is given. In section 3 , solution to the cnnsG equation (1.1 ) is constructed by imposing a constraint on the two basic vectors in double Wronskian. In particular, we focus on the localized soliton solutions and Jordan-block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis. Section 4 is devoted to the conclusions.
2. The nnAKNS equation
We start with the spectral problem [24,25]
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\phi }}}_{x} & =& X\phi ,\quad X=\left(\begin{array}{cc}-\lambda & -u\\ v & \lambda \end{array}\right),\\ & \phi \,=& \left(\begin{array}{l}{\Phi }_{1}\\ {\Phi }_{2}\end{array}\right),\end{array}\end{eqnarray}$
and time evolution $\begin{eqnarray}{\phi }_{t}=Y\phi ,\quad Y=\left(\begin{array}{cc}A & B\\ C & -A\end{array}\right),\end{eqnarray}$
with spectral parameter λ and potentials u=u(x, t) and v=v(x, t). The compatibility condition Φxt=Φtx or moreover zero curvature equation Xt-Yx+[X, Y]=0 leads to $\begin{eqnarray}A={\partial }^{-1}(v,-u)\left(\begin{array}{c}-B\\ C\end{array}\right)-{\lambda }_{t}x+{A}_{0},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\left(\begin{array}{c}-u\\ v\end{array}\right)}_{t}=L\left(\begin{array}{c}-B\\ C\end{array}\right)-2\lambda \left(\begin{array}{c}-B\\ C\end{array}\right)\\ -2{A}_{0}\left(\begin{array}{c}u\\ v\end{array}\right)+2{\lambda }_{t}\left(\begin{array}{c}xu\\ xv\end{array}\right),\end{array}\end{eqnarray}$
in which A0 is a constant and L=σ(∂-2q∂-1qTr), where $q={(-u,v)}^{{\rm{T}}},\sigma =\left(\begin{array}{cc}-1 & 0\\ 0 & 1\end{array}\right),r=\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right)$.Assuming ${A}_{0}=-\frac{1}{2}\gamma (2\lambda )$ and ${\lambda }_{t}=\frac{1}{2}\omega (2\lambda )$ with x-independent Laurent polynomials γ(λ) and ω(λ) and expanding (B, C)T as the Laurent polynomial of λ, one arrives at the general mixed isospectral-nonisospectral AKNS hierarchy [26]2.3 ) appears the isospectral AKNS hierarchy. While when γ(λ)=0 and ω(λ) ≠ 0, then equation (2.3 ) is the pure nonisospectral AKNS hierarchy. In the case of γ(λ) ≠ 0 and ω(λ) ≠ 0, we still refer equation (2.3 ) to as a nonisospectral AKNS hierarchy since the involvement of time-varying spectral parameter λ.
$\begin{eqnarray}{\left(\begin{array}{c}-u\\ v\end{array}\right)}_{t}=\gamma (L)\left(\begin{array}{c}u\\ v\end{array}\right)+\omega (L)\left(\begin{array}{c}xu\\ xv\end{array}\right).\end{eqnarray}$
When γ(λ) ≠ 0 and ω(λ)=0, equation (In order to derive the nnAKNS equation, we set λt=α(t)λ and A0=-(4λ)-1. Taking these expressions into equation (2.3 ) and with straightforward calculations, we get the nnAKNS equation
$\begin{eqnarray}\begin{array}{rcl}{u}_{xt}+2u{\partial }_{x}^{-1}{(uv)}_{t} & =& \alpha (t)[x({u}_{xx}+2{u}^{2}v)\\ & & +2{u}_{x}+2u{\partial }_{x}^{-1}(uv)]+u,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{v}_{xt}+2v{\partial }_{x}^{-1}{(uv)}_{t} & =& \alpha (t)[x({v}_{xx}+2u{v}^{2})\\ & & +2{v}_{x}+2v{\partial }_{x}^{-1}(uv)]+v,\end{array}\end{eqnarray}$
where α(t) is a real function of t. When α(t)=0, it reduces to the usual isospectral negative order AKNS equation [27].The system (2.4 ) can be viewed as a variable-coefficient nnAKNS equation due to the time-dependent coefficient α(t). In recent decades, the studies on the variable-coefficient soliton equations arising from the Lax pair have drawn lots of attention, involving exact solutions and dynamical behaviors [28-32], symmetries [33,34], conservation laws [35,36], and so forth.
Remark 1. Generally speaking, there is a gauge transformation between the first-order nonisospectral and isospectral positive order hierarchies (see [37,38]). Noting that λt=α(t)λ is a linear function of λ, we introduce new coordinate y=e∫α(t)dtx and redefine dependent variables q(y, t)=u(x, t)e-∫α(t)dt, r(y, t)=v(x, t)e-∫α(t)dt. Thus from equation (2.4 ) we know
$\begin{eqnarray}{q}_{yt}+2q{\partial }_{y}^{-1}{(qr)}_{t}={\,\rm{e}\,}^{-\int \alpha (t)\,\rm{d}\,t}q,\end{eqnarray}$
$\begin{eqnarray}{r}_{yt}+2r{\partial }_{y}^{-1}{(qr)}_{t}={\,\rm{e}\,}^{-\int \alpha (t)\,\rm{d}\,t}r,\end{eqnarray}$
which is a variable-coefficient isospectral negative order AKNS equation.To proceed, we introduce the dependent transformation 2.4 ) is transformed into the bilinear equations
$\begin{eqnarray}u=g/f,\quad v=h/f,\end{eqnarray}$
by which equation ( $\begin{eqnarray}({D}_{x}{D}_{t}-\alpha (t)x{D}_{x}^{2})g\cdot f=2\alpha (t){g}_{x}f+gf,\end{eqnarray}$
$\begin{eqnarray}({D}_{x}{D}_{t}-\alpha (t)x{D}_{x}^{2})h\cdot f=2\alpha (t){h}_{x}f+hf,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}f\cdot f=2gh,\end{eqnarray}$
where D is the famous Hirota's bilinear operator defined by [39] $\begin{eqnarray}\begin{array}{rcl} & & {D}_{t}^{m}{D}_{x}^{n}f(t,x)\cdot g(t,x)\\ & & \,={({{\rm{\partial }}}_{t}-{{\rm{\partial }}}_{{t}_{1}})}^{m}\,{({{\rm{\partial }}}_{x}-{{\rm{\partial }}}_{{x}_{1}})}^{n}f(t,x)g(t,x){\left|\right.}_{{t}_{1}=t,{x}_{1}=x}.\end{array}\end{eqnarray}$
For the double Wronskian solutions of bilinear equation (2.7 ), we show the results in the following theorem.
The double Wronski determinants1
$\begin{eqnarray}\begin{array}{rcl}f & =& | \widehat{{\varphi }^{(N)}};\widehat{{\psi }^{(M)}}| ,\\ g & =& 2| \widehat{{\varphi }^{(N+1)}};\widehat{{\psi }^{(M-1)}}| ,\\ & & h=2| \widehat{{\varphi }^{(N-1)}};\widehat{{\psi }^{(M+1)}}| ,\end{array}\end{eqnarray}$
solve the bilinear equation ( $\begin{eqnarray}{\varphi }_{x}={\rm{\Lambda }}(t)\varphi ,\quad {\psi }_{x}=-{\rm{\Lambda }}(t)\psi ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\varphi }_{t}=\frac{1}{4}{\partial }^{-1}\varphi +\alpha (t)(x{\varphi }_{x}-N\varphi ),\\ {\psi }_{t}=\frac{1}{4}{\partial }^{-1}\psi +\alpha (t)(x{\psi }_{x}-M\psi ),\end{array}\end{eqnarray}$
where λ(t) is a (N + M + 2) × (N + M + 2) invertible matrix of t but independent of x.The CES equation (2.10 ) implies 2.4 ) can be derived.
$\begin{eqnarray}\begin{array}{c}\varphi ={\rm{e}}^{{\rm{\Lambda }}(t)x+\frac{1}{4}\displaystyle \int ({{\rm{\Lambda }}}^{-1}(t)-4N\alpha (t)I)\rm{d}t}{C}^{+},\\ \psi ={\rm{e}}^{-{\rm{\Lambda }}(t)x-\frac{1}{4}\displaystyle \int ({{\rm{\Lambda }}}^{-1}(t)+4M\alpha (t)I)\rm{d}t}{C}^{-},\end{array}\end{eqnarray}$
with constraint λt(t)=α(t)λ(t), where C± are constant column vectors. Here and hereafter I is the unit matrix whose index indicating the size is omitted. In terms of different eigenvalue structures of matrix λ(t), various exact solutions, including soliton solutions, Jordan-block solutions, rational solutions, and mixed solutions, for the nnAKNS system (In the following section, we would like to derive the cnnsG equation. For the resulting equation, we shall introduce some conditions on λ(t), φ and ψ to construct the localized solutions. To this end, we take M=N, under which the parts ∫(Nα(t)I)dt and ∫(Mα(t)I)dt in equation (2.11 ) do not contribute to u or v. Thus we omit them in the construction of solutions.
3. The cnnsG equation and localized wave
We impose v=u* on the system (2.4 ) and get the cnnsG equation (1.1 ), i.e.
$\begin{eqnarray}\begin{array}{rcl}{u}_{xt}+2u{\partial }_{x}^{-1}{(u{u}^{* })}_{t} & =& \alpha (t)[x({u}_{xx}+2{u}^{2}{u}^{* })\\ & & +2{u}_{x}+2u{\partial }_{x}^{-1}(u{u}^{* })]+u.\end{array}\end{eqnarray}$
The double Wronskian solutions of equation (1.1 ) can be formulated as follows.
The function u=g/f with
$\begin{eqnarray}f=| \widehat{{\varphi }^{(N)}};\widehat{{\psi }^{(N)}}| ,\quad g=2| \widehat{{\varphi }^{(N+1)}};\widehat{{\psi }^{(N-1)}}| ,\end{eqnarray}$
solve the cnnsG equation ( $\begin{eqnarray}\psi =T{\varphi }^{* },\end{eqnarray}$
where $T\in {{\mathbb{C}}}^{(2N+2)\times (2N+2)}$ is a constant matrix satisfying $\begin{eqnarray}{\rm{\Lambda }}(t)T+T{{\rm{\Lambda }}}^{\ast }(t)={\bf{0}},\,T{T}^{\ast }=-I,\end{eqnarray}$
and ${C}^{-}=T{C}^{{+}^{* }}$.Under the assumption equation (
$\begin{eqnarray}f(x)=| {\widehat{\phi }}^{(N)};{\widehat{\psi }}^{(N)}| =| {\widehat{\phi }}^{(N)};T{\widehat{\phi }}^{* (N)}| ,\end{eqnarray}$
$\begin{eqnarray}g(x)=2| {\widehat{\phi }}^{(N+1)};{\widehat{\psi }}^{(N-1)}| =2| {\widehat{\phi }}^{(N+1)};T{\widehat{\phi }}^{* (N-1)}| ,\end{eqnarray}$
$\begin{eqnarray}h(x)=2| {\widehat{\phi }}^{(N-1)};{\widehat{\psi }}^{(N+1)}| =2| {\widehat{\phi }}^{(N-1)};T{\widehat{\phi }}^{* (N+1)}| .\end{eqnarray}$
Then we rearrange f as
$\begin{eqnarray}\begin{array}{rcl}{f}^{* } & =& | {\widehat{\phi }}^{(N)};T{\widehat{\phi }}^{* (N)}{| }^{* }=| {\widehat{\phi }}^{* (N)};{T}^{* }{\widehat{\phi }}^{(N)}| \\ & =& | {T}^{* }| | -T{\widehat{\phi }}^{* (N)};{\widehat{\phi }}^{(N)}| =| {T}^{* }| {(-1)}^{{(N+1)}^{2}}| {\widehat{\phi }}^{(N)};-T{\widehat{\phi }}^{* (N)}| \\ & =& | {T}^{* }| | {\widehat{\phi }}^{(N)};T{\widehat{\phi }}^{* (N)}| =| {T}^{* }| f,\end{array}\end{eqnarray}$
and similarly g*=∣T*∣h, which give rise to $\begin{eqnarray}{u}^{* }={g}^{* }/{f}^{* }=(| {T}^{* }| h)/(| {T}^{* }| f)=h/f=v.\end{eqnarray}$
Thus we finish the proof. The first equation encoded in equation (3.4 ) is often referred to as the Sylvester equation (see [40,41]), usually appearing in many contexts of integrable systems (see [42] and the references therein). The above theorem tells us that u=g/f with 1.1 ), in which $\varphi ={({\varphi }^{+},{\varphi }^{-})}^{{\rm{T}}}\,={\,\rm{e}\,}^{{\rm{\Lambda }}(t)x+\frac{1}{4}\int {{\rm{\Lambda }}}^{-1}(t)\,\rm{d}\,t}{C}^{+}$ and T and λ(t) satisfy the matrix equation (3.4 ), where ${\varphi }^{\pm }=({\varphi }_{1}^{\pm },{\varphi }_{2}^{\pm },\ldots ,{\varphi }_{N+1}^{\pm })$.
$\begin{eqnarray}f=| \widehat{{\varphi }^{(N)}};T\widehat{{\varphi }^{{(N)}^{* }}}| ,\quad g=2| \widehat{{\varphi }^{(N+1)}};T\widehat{{\varphi }^{{(N-1)}^{* }}}| \end{eqnarray}$
solves the cnnsG equation (To present the explicit solutions of cnnsG equation (1.1 ), we take 1.1 ) are independent of phase parameters C+, i.e., the initial phase has always to be 0. According to the equation (3.6 ), we know that f is a real function in light of ∣T∣=1.
$\begin{eqnarray}\begin{array}{c}{\rm{\Lambda }}(t)=\left(\begin{array}{cc}K(t) & {\bf{0}}\\ {\bf{0}} & -{K}^{\ast }(t)\end{array}\right),\,T=\left(\begin{array}{cc}{\bf{0}} & I\\ -I & {\bf{0}}\end{array}\right),\end{array}\end{eqnarray}$
where K(t) is a (N+1) × (N+1) invertible matrix satisfying Kt(t)=α(t)K(t). Due to the block structure of matrix T, we note that C+ can be gauged to be $\bar{I}={(1,1,\ldots ,1;1,1,\ldots ,1)}^{{\rm{T}}}$ or $\breve{I}={(1,0,\ldots ,0;1,0,\ldots ,0)}^{{\rm{T}}}$. It implies that the solutions for equation (First note that since the solutions are given in terms of determinants, any matrix similar to K(t) yields same solution as K(t). Thus, it is sufficient to consider only different canonical forms of K(t) (see [43,44]). For the sake of brevity, we introduce some notations
$\begin{eqnarray}\begin{array}{l}\kappa (t)={\,\rm{e}\,}^{\displaystyle \int \alpha (t)\,\rm{d}\,t},\quad {k}_{j}(t)={l}_{j}\kappa (t),\\ {\xi }_{j}=2{l}_{j}\kappa (t)x+\frac{1}{2{l}_{j}}\displaystyle \int {\kappa }^{-1}(t)\,\rm{d}\,t,\end{array}\end{eqnarray}$
where lj, (j=1, 2, …, N+1) are complex constants.Case 1. If K(t) is a diagonal matrix,
$\begin{eqnarray}K(t)=\,\rm{diag}\,({k}_{1}(t),{k}_{2}(t),\ldots ,{k}_{N+1}(t)),\end{eqnarray}$
where complex constants {lj} distinguish the spectral parameters {kj(t)}, then the entries in the Wronskian can be taken as $\begin{eqnarray}{\varphi }_{j}^{+}={\,\rm{e}\,}^{{\xi }_{j}},\quad {\varphi }_{j}^{-}={\,\rm{e}\,}^{-{\xi }_{j}^{* }},\quad j=1,2,\ldots ,N+1,\end{eqnarray}$
where we have taken ${C}^{+}=\bar{I}$. This case leads to the soliton solutions.Case 2. Next suppose that K(t) is a lower triangular matrix defined as 3.12 ). The solutions obtained in this case are usually referred to as Jordan-block (or limit) solutions (see [43,44]).
$\begin{eqnarray}K(t)={\left(\begin{array}{ccccc}{k}_{1}(t) & 0 & \ldots & 0 & 0\\ \kappa (t) & {k}_{1}(t) & \ldots & 0 & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & \kappa (t) & {k}_{1}(t)\end{array}\right)}_{(N+1)\times (N+1)},\end{eqnarray}$
and ${C}^{+}=\breve{I}$. In this case, we know that $\begin{eqnarray}{\varphi }_{j}^{+}=\frac{{\partial }_{{l}_{1}}^{j-1}{\varphi }_{1}^{+}}{(j-1)!},\quad {\varphi }_{j}^{-}=\frac{{\partial }_{{l}_{1}^{* }}^{j-1}{\varphi }_{1}^{-}}{(j-1)!},\quad j=2,\ldots ,N+1,\end{eqnarray}$
where ${\varphi }_{1}^{\pm }$ are given by equation (Function κ(t) (or α(t)) is a crucial factor to show the localized waves of the above solutions. In what follows, we consider κ(t) as a Gaussian-variant-type function of t such as $\kappa (t)={\rm{sech}}\, t$ or $\kappa (t)=\frac{1}{{t}^{2}+1}$. Without loss of generality, we take $\kappa (t)={\rm{sech}}\, t$ (or $\alpha (t)={(\mathrm{ln}{\rm{sech}}\, t)}_{t}$).
In the case of N=0, the one-soliton solution reads
$\begin{eqnarray}u=\displaystyle \frac{4(\,\rm{Re}\,{l}_{1}){\rm{sech}}\, t\cdot \exp [2{l}_{1}x\,{\rm{sech}}\, t+(\sinh\,t)/(2{l}_{1})]}{1+\exp \left[2(\,\rm{Re}\,{l}_{1})(4| {l}_{1}{| }^{2}x\,{\rm{sech}}\, t+\sinh\,t)/(2| {l}_{1}{| }^{2})\right]},\end{eqnarray}$
where and whereafter Re(·) means the real part of complex number and ∣·∣ denotes modulus.To consider the dynamics of this solution, we denote l1=(a1+ib1)/2 and get 3.16 ) is depicted in figure 1.
$\begin{eqnarray}\begin{array}{l}| u{| }^{2}={a}_{1}^{2}{{\rm{sech}} }^{2}t\cdot {{\rm{sech}} }^{2}{\eta }_{1},\quad \,\rm{with}\,\\ {\eta }_{1}={a}_{1}x\,{\rm{sech}}\, t+{a}_{1}(\sinh\,t)/({a}_{1}^{2}+{b}_{1}^{2}),\end{array}\end{eqnarray}$
which is a localized wave, since the term $4{a}_{1}^{2}{{\rm{sech}} }^{2}t$ provides a Gaussian t-variant amplitude. This wave travels with hyperbolic sine type top trace $-\sinh 2t/(2({a}_{1}^{2}+{b}_{1}^{2}))$, and the hyperbolic cosine type velocity $-\cosh 2t/({a}_{1}^{2}+{b}_{1}^{2})$. It is clear that the amplitude is controlled by one parameter a1. While both parameters a1 and b1 together govern the top trace and velocity. Solution (
Figure 1. One-soliton solution ∣u∣2 given by equation ( |
In the case of N=1, the two-soliton solutions are of form ∣u∣2=gg*/f2, where
$\begin{eqnarray}\begin{array}{rcl}f & =& (| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2})(1+{\rm{e}\,}^{2\,\rm{Re}{\xi }_{1}})(1+{\rm{e}\,}^{2\,\rm{Re}{\xi }_{2}})\\ & & -2\,\rm{Re}\,({l}_{1}{l}_{2}^{* })| 1+{\,\rm{e}\,}^{{\xi }_{1}+{\xi }_{2}^{* }}{| }^{2}\\ & & +2\,\rm{Re}\,({l}_{1}{l}_{2})| {\rm{e}\,}^{{\xi }_{1}}-{\,\rm{e}}^{{\xi }_{2}}{| }^{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}g & =& 4{\rm{sech}}\, t\left\{({l}_{1}^{* }-{l}_{2}^{* })[({l}_{1}+{l}_{2}^{* })(\,\rm{Re}\,{l}_{1}){\,\rm{e}\,}^{{\xi }_{1}}\right.\\ & & -({l}_{1}^{* }+{l}_{2})(\,\rm{Re}\,{l}_{2}){\,\rm{e}\,}^{{\xi }_{2}}]\\ & & -({l}_{1}-{l}_{2})[({l}_{1}+{l}_{2}^{* })(\,\rm{Re}\,{l}_{2}){\,\rm{e}\,}^{{\xi }_{1}^{* }}\\ & & \left.-({l}_{1}^{* }+{l}_{2})(\,\rm{Re}\,{l}_{1}){\,\rm{e}\,}^{{\xi }_{2}^{* }}]{\,\rm{e}\,}^{{\xi }_{1}+{\xi }_{2}}\right\}.\end{array}\end{eqnarray}$
Now, let us analyze the interaction of two-soliton. For convenience, the asymptotic solitons are called as k1-soliton and k2-soliton, respectively. We rewrite ${\xi }_{j}={{ \mathcal A }}_{j}+{\rm{i}}{{ \mathcal B }}_{j},(j=1,2)$ with
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{j} & =& {a}_{j}(x\,{\rm{sech}}\, t+{({a}_{j}^{2}+{b}_{j}^{2})}^{-1}\sinh\,t)\quad \,\rm{and}\,\\ {{ \mathcal B }}_{j} & =& {b}_{j}(x\,{\rm{sech}}\, t-{({a}_{j}^{2}+{b}_{j}^{2})}^{-1}\sinh\,t).\end{array}\end{eqnarray}$
It is easy to find that ${\,\rm{e}\,}^{{\xi }_{j}}\to \infty $ as ${{ \mathcal A }}_{j}\to +\infty $ and ${\,\rm{e}\,}^{{\xi }_{j}}\to 0$ as ${{ \mathcal A }}_{j}\to -\infty $. Without loss of generality, we assume 0 < a2 < a1 and 0 < b2 < b1 to guarantee ${ \mathcal X }={({a}_{2}^{2}+{b}_{2}^{2})}^{-1}-{({a}_{1}^{2}+{b}_{1}^{2})}^{-1}\gt 0$.
For fixed ξ1, it follows from equation (3.10 ) that 3.21 ) and (3.23 ) can be expressed as
$\begin{eqnarray}{\xi }_{2}={a}_{2}{\xi }_{1}/{a}_{1}+{a}_{2}{ \mathcal X }\sinh\,t+{\rm{i}}({a}_{1}{{ \mathcal B }}_{2}-{a}_{2}{{ \mathcal B }}_{1})/{a}_{1},\end{eqnarray}$
satisfies ${\,\rm{e}\,}^{{\xi }_{2}}\to \infty $ as t →+∞ and ${\,\rm{e}\,}^{{\xi }_{2}}\to 0$ as t →-∞. When t →+∞, neglecting subdominant exponential terms, we have $\begin{eqnarray}f\simeq {\,\rm{e}}^{2\rm{Re}\,{\xi }_{2}}[(| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2})(1+{\,\rm{e}}^{2\rm{Re}\,{\xi }_{1}})-2\,\rm{Re}\,({l}_{1}{l}_{2}^{* }){\,\rm{e}}^{2\rm{Re}{\xi }_{1}}+2\rm{Re}\,({l}_{1}{l}_{2})],\end{eqnarray}$
$\begin{eqnarray}g\simeq 4(\,\rm{Re}\,{l}_{1})({l}_{1}-{l}_{2})({l}_{1}^{* }+{l}_{2}){\rm{e}\,}^{{\xi }_{1}+2\,\rm{Re}{\xi }_{2}}{\rm{sech}}\, t,\end{eqnarray}$
and thus the k1-soliton appears $\begin{eqnarray}u\simeq \frac{4(\,\rm{Re}\,{l}_{1})({l}_{1}-{l}_{2})({l}_{1}^{* }+{l}_{2}){\rm{e}\,}^{{\xi }_{1}}{\rm{sech}}\, t}{| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2}+2\,\rm{Re}({l}_{1}{l}_{2})+(| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2}-2\,\rm{Re}\,({l}_{1}{l}_{2}^{* })){\rm{e}\,}^{2\,\rm{Re}{\xi }_{1}}}.\end{eqnarray}$
When t →-∞, we find $\begin{eqnarray}f\simeq (| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2})(1+{\,\rm{e}}^{2\rm{Re}\,{\xi }_{1}})-2\,\rm{Re}\,({l}_{1}{l}_{2}^{* })+2\,\rm{Re}\,({l}_{1}{l}_{2}){\,\rm{e}}^{2\rm{Re}\,{\xi }_{1}},\end{eqnarray}$
$\begin{eqnarray}g\simeq 4(\,\rm{Re}\,{l}_{1})({l}_{1}^{* }-{l}_{2}^{* })({l}_{1}+{l}_{2}^{* }){\,\rm{e}\,}^{{\xi }_{1}}{\rm{sech}}\, t,\end{eqnarray}$
and thus the k1-soliton exhibits $\begin{eqnarray}u\simeq \frac{4(\,\rm{Re}\,{l}_{1})({l}_{1}^{* }-{l}_{2}^{* })({l}_{1}+{l}_{2}^{* }){\rm{e}\,}^{{\xi }_{1}}{\rm{sech}}\, t}{| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2}-2\,\rm{Re}({l}_{1}{l}_{2}^{* })+(| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2}+2\,\rm{Re}\,({l}_{1}{l}_{2})){\rm{e}\,}^{2\,\rm{Re}{\xi }_{1}}}.\end{eqnarray}$
As a results, the envelops of equations ( $\begin{eqnarray}| u{| }^{2}\simeq {a}_{1}^{2}{{\rm{sech}} }^{2}t\cdot {{\rm{sech}} }^{2}\left({\eta }_{1}\pm \frac{1}{2}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}}\right),\quad t\to \pm \infty ,\end{eqnarray}$
where ${\cdot }_{{ij}}^{\pm }{\,=\,\cdot }_{i}{\pm \cdot }_{j}$, which travel with Gaussian t-variant amplitude ${a}_{1}^{2}{{\rm{sech}} }^{2}t$, trajectory $\begin{eqnarray}x(t)=-\frac{\sinh 2t}{2({a}_{1}^{2}+{b}_{1}^{2})}\mp \frac{\cosh t}{2{a}_{1}}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}},\quad t\to \pm \infty ,\end{eqnarray}$
and velocity $\begin{eqnarray}{x}^{{\prime} }(t)=-\frac{\cosh 2t}{{a}_{1}^{2}+{b}_{1}^{2}}\mp \frac{\sinh\,t}{2{a}_{1}}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}},\quad t\to \pm \infty .\end{eqnarray}$
The phase shift is consequently expressed as $\begin{eqnarray}-\frac{\cosh t}{{a}_{1}}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}}.\end{eqnarray}$
For fixed ξ2, it follows from equation (3.10 ) that 3.30 ) and (3.32 ) are
$\begin{eqnarray}{\xi }_{1}={a}_{1}{\xi }_{2}/{a}_{2}-{a}_{1}{ \mathcal X }\sinh\,t+i({a}_{2}{{ \mathcal B }}_{1}-{a}_{1}{{ \mathcal B }}_{2})/{a}_{2},\end{eqnarray}$
satisfies ${\,\rm{e}\,}^{{\xi }_{1}}\to 0$ as t →+∞ and ${\,\rm{e}\,}^{{\xi }_{1}}\to \infty $ as t →-∞. When t →+∞, neglecting subdominant exponential terms we derive $\begin{eqnarray}\begin{array}{rcl}f & \simeq & (| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2})(1+{\rm{e}\,}^{2\,\rm{Re}{\xi }_{2}})-2\,\rm{Re}\,({l}_{1}^{* }{l}_{2})\\ & & +2\,\rm{Re}\,({l}_{1}{l}_{2}){\rm{e}\,}^{2\,\rm{Re}{\xi }_{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}g\simeq -4(\,\rm{Re}\,{l}_{2})({l}_{1}^{* }-{l}_{2}^{* })({l}_{1}^{* }+{l}_{2}){\,\rm{e}\,}^{{\xi }_{2}}{\rm{sech}}\, t,\end{eqnarray}$
the k2-soliton thereby reads as $\begin{eqnarray}u\simeq -\frac{4(\,\rm{Re}\,{l}_{2})({l}_{1}^{* }-{l}_{2}^{* })({l}_{1}^{* }+{l}_{2}){\,\rm{e}\,}^{{\xi }_{2}}{\rm{sech}}\, t}{(| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2})(1+{\rm{e}\,}^{2\,\rm{Re}{\xi }_{2}})-2\,\rm{Re}\,({l}_{1}^{* }{l}_{2})+2\,\rm{Re}\,({l}_{1}{l}_{2}){\rm{e}\,}^{2\,\rm{Re}{\xi }_{2}}}.\end{eqnarray}$
When t →-∞, since $\begin{eqnarray}f\simeq {\rm{e}\,}^{2\,\rm{Re}{\xi }_{1}}[(| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2})(1+{\rm{e}\,}^{2\,\rm{Re}{\xi }_{2}})-2\,\rm{Re}\,({l}_{1}^{* }{l}_{2}){\rm{e}\,}^{2\rm{Re}{\xi }_{2}}+2\,\rm{Re}({l}_{1}{l}_{2})],\end{eqnarray}$
$\begin{eqnarray}g\simeq -4(\,\rm{Re}\,{l}_{2})({l}_{1}-{l}_{2})({l}_{1}+{l}_{2}^{* }){\rm{e}\,}^{{\xi }_{2}+2\,\rm{Re}{\xi }_{1}}{\rm{sech}}\, t,\end{eqnarray}$
the k2-soliton thereby becomes $\begin{eqnarray}u\simeq -\frac{4(\,\rm{Re}\,{l}_{2})({l}_{1}-{l}_{2})({l}_{1}+{l}_{2}^{* }){\,\rm{e}\,}^{{\xi }_{2}}{\rm{sech}}\, t}{(| {l}_{1}{| }^{2}+| {l}_{2}{| }^{2})(1+{\rm{e}\,}^{2\,\rm{Re}{\xi }_{2}})-2\,\rm{Re}\,({l}_{1}^{* }{l}_{2}){\rm{e}\,}^{2\rm{Re}{\xi }_{2}}+2\,\rm{Re}({l}_{1}{l}_{2})}.\end{eqnarray}$
The envelops of k2-soliton equations ( $\begin{eqnarray}| u{| }^{2}\simeq {a}_{2}^{2}{{\rm{sech}} }^{2}t\cdot {{\rm{sech}} }^{2}\left({\eta }_{2}\mp \frac{1}{2}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}}\right),\quad t\to \pm \infty ,\end{eqnarray}$
with ${\eta }_{2}={\eta }_{1}{| }_{{a}_{1}\to {a}_{2},{b}_{1}\to {b}_{2}}$, which travel with Gaussian t-variant amplitude ${a}_{2}^{2}{{\rm{sech}} }^{2}t$, trajectory $\begin{eqnarray}x(t)=-\frac{\sinh 2t}{2({a}_{2}^{2}+{b}_{2}^{2})}\pm \frac{\cosh t}{2{a}_{2}}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}},\quad t\to \pm \infty ,\end{eqnarray}$
and velocity $\begin{eqnarray}{x}^{{\prime} }(t)=-\frac{\cosh 2t}{{a}_{2}^{2}+{b}_{2}^{2}}\pm \frac{\sinh\,t}{2{a}_{2}}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}},\quad t\to \pm \infty .\end{eqnarray}$
The phase shift is consequently expressed as $\begin{eqnarray}\frac{\cosh t}{{a}_{2}}\mathrm{ln}\frac{{a}_{12}^{{-}^{2}}+{b}_{12}^{{-}^{2}}}{{a}_{12}^{{+}^{2}}+{b}_{12}^{{-}^{2}}}.\end{eqnarray}$
We depict the solution (3.17 ) in figure 2.
Figure 2. Two-soliton solutions ∣u∣2 given by equation ( |
The simplest Jordan-block solution reads ∣u∣2=gg*/f2 with
$\begin{eqnarray}\begin{array}{rcl}f & =& (\,\rm{Re}\,{l}_{1}){\left|4{l}_{1}x\,{\rm{sech}}\, t-(\sinh\,t)/{l}_{1}\right|}^{2}+2| {l}_{1}{| }^{2}\\ & & \times (1+\cosh (2\,\rm{Re}\,{\xi }_{1})),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}g & =& 2{\rm{sech}}\, t\left\{{l}_{1}[4{l}_{1}^{* }+2(\,\rm{Re}\,{l}_{1})(4{l}_{1}x\,{\rm{sech}}\, t-(\sinh\,t)/{l}_{1}^{* })]{\rm{e}\,}^{2\,\rm{Re}{\xi }_{1}}\right.\\ & & \left.-{l}_{1}^{* }\left[({l}_{1}^{* }/{l}_{1})\sinh\,t-{l}_{1}(4+8(\,\rm{Re}\,{l}_{1})x\,{\rm{sech}}\, t-(\sinh\,t)/{l}_{1})\right]\right\}{\,\rm{e}\,}^{-{\xi }_{1}^{* }}.\end{array}\end{eqnarray}$
This solution is sketched in figure 3.
Figure 3. Jordan-block solution ∣u∣2 given by (3.37) with l1=0.15+i: (a) shape and motion; (b) Density plot of (a) with larger range x ∈ [-200, 200], t ∈ [-5, 5]; (c) 2D-plot of (a) at t=0. |
Figure 4 illustrates the interaction of between one-soliton and the simplest Jordan-block solution, where ∣u∣2=gg*/f2 with
$\begin{eqnarray}f=| {\phi }^{(0)},{\phi }^{(1)},{\phi }^{(2)};{\psi }^{(0)},{\psi }^{(1)},{\psi }^{(2)}| ,\end{eqnarray}$
$\begin{eqnarray}g=2| {\phi }^{(0)},{\phi }^{(1)},{\phi }^{(2)},{\phi }^{(3)};{\psi }^{(0)},{\psi }^{(1)}| ,\end{eqnarray}$
where $\begin{eqnarray}{\phi }^{(0)}={({\rm{e}\,}^{{\xi }_{1}},{\rm{e}}^{{\xi }_{2}},{\partial }_{{l}_{2}}{\rm{e}}^{{\xi }_{2}},{\rm{e}}^{-{\xi }_{1}^{* }},{\rm{e}}^{-{\xi }_{2}^{* }},{\partial }_{{l}_{2}^{* }}{\,\rm{e}}^{-{\xi }_{2}^{* }})}^{T},\end{eqnarray}$
$\begin{eqnarray}{\psi }^{(0)}={({\rm{e}\,}^{-{\xi }_{1}},{\rm{e}}^{-{\xi }_{2}},{\partial }_{{l}_{2}}{\rm{e}}^{-{\xi }_{2}},-{\rm{e}}^{{\xi }_{1}^{* }},-{\rm{e}}^{{\xi }_{2}^{* }},-{\partial }_{{l}_{2}^{* }}{\,\rm{e}}^{{\xi }_{2}^{* }})}^{T}.\end{eqnarray}$
Figure 4. Interactions of one-soliton and the simplest Jordan-block solution given by equation ( |
4. Conclusions
The localized wave solutions, in contrast to their continuous wave or monochromatic counterparts, exhibit enhanced localization and energy fluence characteristics. Since the fact that a time-dependent spectral parameter may produce soliton solutions with localized amplitude with respect to time, in this paper we study localized waves to the cnnsG equation (1.1 ), which is a complex reduction equation of the nnAKNS equation (2.4 ). In the cnnsG equation (1.1 ) (or nnAKNS equation (2.4 )), α(t) is a real function of t, which exerts important roles in the generation of localized waves. We use the bilinearization reduction method to construct solutions of the cnnsG equation (1.1 ). In terms of the different eigenvalue structures of matrix K(t), formal soliton solutions and Jordan-block solutions are presented. To exhibit the localized waves of the resulting solutions, we take $\alpha (t)={(\mathrm{ln}{\rm{sech}}\, t)}_{t}$. As a consequence, the one-, two-soliton solutions, the simplest Jordan-block solution as well as the simplest mixed solution are given as four examples of the localized solutions. Dynamics of these solutions are specially analyzed and illustrated.
We finish the paper by the following remarks.
First of all, comparing with the usual nonisospectral soliton equations with time-varying amplitudes (not space-time localized waves) [45-48], the introduction of $\alpha (t)={(\mathrm{ln}{\rm{sech}}\, t)}_{t}$ by λt=α(t)λ leads to the Gaussian-variant-type amplitudes (space-time localized waves) of the objective nonisospectral soliton equations [20]. This idea is based on the fact that in isospectral case amplitude of a soliton is usually governed by the spectral parameter and then introducing a time-dependent spectral parameter may generate a localized amplitude with respect to time.
What is more, there are a variety of studies on the gauge transformations between the first-order nonisospectral and isospectral positive order hierarchies [37,38]. While, the results on gauge transformations between the first-order nonisospectral and isospectral negative order hierarchies are rather sparse. By introducing new coordinate y and new dependent variables q(y, t) and r(y, t), we transform the nnAKNS equation (2.4 ) into a isospectral form (see remark 1), whereas it appears with variable-coefficient without constant-coefficient. It is undoubtedly that figuring out the gauge transformation theory between the first-order nonisospectral and isospectral negative order hierarchies is meaningful and worth considering.
In addition, there are several physically important negative order integrable systems, such as the Fokas-Lenells equation [49,50] and massive Thirring model [51,52]. A natural question is how we construct their first-order nonisospectral correspondences and study the localized waves generated by α(t). Besides, can the research ideas and methods in this paper be generalized to investigate the localized waves of semi-discrete integrable systems? These are the intriguing questions deserved further consideration, which will done in near future.
Declaration of competing interest
The authors declare that there are no conflicts of interests with publication of this work.
Author contributions
SLZ: formal analysis, methodology, validation, supervision, writing-review & editing. XHF: investigation, writing-original draft, validation, software, validation.
Acknowledgments
The authors are very grateful to the referees for their invaluable comments. This project is supported by the National Natural Science Foundation of China (Grant No. 12071432) and Zhejiang Provincial Natural Science Foundation (Grant No. LZ24A010007).