1. Introduction
As a nonlinear Schrödinger (NLS) type equation, the Gross–Pitaevskii (GP) equation [1–3],1 ), is a nonisospectral NLS equation that is integrable and related to a time-dependent spectral parameter λ = t, and can be gauge transformed to the usual NLS equation [5]. When V ∝ x2, usually the GP equation with constant coefficients, is not integrable, but there are some integrable cases when time-dependent coefficients are introduced (e.g. [6–9]).
$\begin{eqnarray}{\rm{i}}{\rm{\hslash }}\displaystyle \frac{\partial \psi (\vec{r})}{\partial t}=\left(-\displaystyle \frac{{{\rm{\hslash }}}^{2}}{2m}{{\rm{\nabla }}}^{2}+V(\vec{r})+g| \psi (\vec{r}){| }^{2}\right)\psi (\vec{r}),\end{eqnarray}$
describes the ground state of a quantum system of identical bosons, and in particular, the Bose–Einstein condensates (BEC) with a constant number of atoms at temperatures close to absolute zero (see [4] and the references therein). In the equation, $\psi (\vec{r})$ is the wave function in three-dimensional space, $V(\vec{r})$ is the external potential, i, ℏ, m and g stand for the imaginary unit, the reduced Planck constant, the mass of the bosons, and a parameter that measures inter-particle interactions, respectively. When V ∝ x, the one-dimensional (1D) dimensionless version of the GP equation (Recently, the authors considered a dimensionless GP equation with a parabolic external potential of the following form [10],2 ) exhibit interesting interaction dynamics but ∣q∣2 is not a conserved density.
$\begin{eqnarray}{\rm{i}}{q}_{t}+{q}_{{xx}}+2| q{| }^{2}q+(\delta {x}^{2}+{\rm{i}}\alpha (t))q=0\end{eqnarray}$
with δ being a real constant. This equation is integrable when α = α(t) is a real function of t and governed by $\begin{eqnarray}{\alpha }_{t}+2{\alpha }^{2}=2\delta .\end{eqnarray}$
Note that there is a gain term iα(t)q in the equation, which means one needs to either add or reduce atoms in experiments, and thus the number of atoms is not constant. Solutions of (In this paper, we will consider the following GP equation
$\begin{eqnarray}{\rm{i}}{u}_{t}+{u}_{{xx}}+2g(t)| u{| }^{2}u+\delta {x}^{2}u=0,\end{eqnarray}$
where $\begin{eqnarray}g(t)=\left\{\begin{array}{ll}{\rm{sech}} (2\sqrt{\delta }t), & \delta \gt 0,\\ \sec (2\sqrt{-\delta }t), & \delta \lt 0.\end{array}\right.\end{eqnarray}$
This equation can be derived from (2 ) by analyzing a one-soliton solution and the corresponding quantity ${\int }_{-\infty }^{+\infty }| q{| }^{2}{\rm{d}}x$ of the equation (2 ). Details will be explained in the next section. Note that g(t) plays a role that measures inter-particle interactions by formally replacing the carrier wave ∣u∣2 in the constant-coefficient GP equation using g(t)∣u∣2. We will see that the equation (4 ) is not only integrable but also allows the conserved particle density ∣u∣2. Its infinitely many conservation laws and multi-soliton solutions will be derived. As for dynamics, the carrier wave ∣u∣2 does not have a constant amplitude although ${\int }_{-\infty }^{+\infty }| u{| }^{2}{\rm{d}}x$ is a conserved quantity. ∣u∣2 is a localized-like wave when δ > 0, but has periodic singularities when δ < 0. We also note that equation (4a ) together with parametrisation (4b ) was already considered in the literature, e.g. in [7], where a more general form of the GP equation with time-dependent coefficients was investigated and equation (4a ) is one of the cases that are gauge equivalent to the NLS equation with constant coefficients (see table 1 in [7]). However, in our paper, we focus on how a conserved GP equation is obtained from a non-conserved one by observing a soliton solution.
We will also extend our idea from the conserved GP equation (4 ) to the vector case, i.e.4b ) and measures the atomic interactions, δx2 is an external parabolic potential and † stands for the Hermitian of a matrix. A vector GP system can be used to describe the dynamics of multi-interacting dilute Bose condensates at absolute zero temperature [11–14]. This system is integrable and related to nonisospectral vector NLS equations. Infinitely many conservation laws of (5 ) will be constructed, which show that the particle density ${\sum }_{j=1}^{n}| {u}_{j}{| }^{2}$ is conserved. We will also derive a gauge transformation that connects the conserved GP equation (5 ) to an isospectral vector NLS equation. Moreover, N-soliton solutions (NSS) will be constructed via gauge transformation, and the dynamics of some solutions will be analyzed and illustrated.
$\begin{eqnarray}{\rm{i}}{{\boldsymbol{u}}}_{t}+{{\boldsymbol{u}}}_{{xx}}+2g(t)({{\boldsymbol{u}}}^{\dagger }{\boldsymbol{u}}){\boldsymbol{u}}+\delta {x}^{2}{\boldsymbol{u}}=0,\end{eqnarray}$
where (referring to [11]) ${\boldsymbol{u}}={({u}_{1},\cdots ,{u}_{n})}^{{\rm{T}}}$ is an n-component complex-valued vector field with ui(i = 1, 2, ⋯ ,n) being the wave function of the i-th Bose condensate, g(t) takes the form (The paper is organized as follows. In section 2 , we explain the main idea of how we construct the conserved GP equation (4 ) from a non-conserved GP equation, based on solutions of the latter. In section 3 , conserved vector GP equation (5 ) are investigated, including a Lax pair, infinitely many conservation laws, gauge transformation, and NSS, as well as their dynamics. Finally, section 4 serves for conclusions.
2. The conserved GP equation (4 )
In this section, we investigate the conserved GP equation (4 ) and its integrability.
Let us first explain how we are motivated by solutions of the nonconserved GP equation (2 ) and arrive at the conserved equation (4 ). Recalling the carrier wave (particle density) of the explicit 1SS of (2 ) [10],3 ) is
$\begin{eqnarray}\begin{array}{rcl}| q{| }^{2} & = & 4{a}_{1}^{2}{{\rm{sech}} }^{2}(2\sqrt{\delta }t)\,{{\rm{sech}} }^{2}\left[2{a}_{1}{\rm{sech}} (2\sqrt{\delta }t)\right.\\ & & \left.\times \left(x+\displaystyle \frac{2{b}_{1}}{\sqrt{\delta }}\sinh (2\sqrt{\delta }t)-{h}_{1}\cosh (2\sqrt{\delta }t)\right)\right],\end{array}\end{eqnarray}$
where ${a}_{1},{b}_{1},{h}_{1}\in {\mathbb{R}}$ are arbitrary, which provides a space-time localized wave when δ > 0. In order to understand how the gain term affects the total particle number, by calculation, we find the number is $\begin{eqnarray}{ \mathcal N }={\int }_{-\infty }^{\infty }| q{| }^{2}{\rm{d}}x=4| {a}_{1}| {\rm{sech}} (2\sqrt{\delta }t).\end{eqnarray}$
Note that the function α(t) satisfying ( $\begin{eqnarray}\alpha (t)=\left\{\begin{array}{ll}\sqrt{\delta }\tanh (2\sqrt{\delta }t), & \delta \gt 0,\\ -\sqrt{-\delta }\tan (2\sqrt{-\delta }t), & \delta \lt 0.\end{array}\right.\end{eqnarray}$
This indicates how the total number of atoms is affected by α(t).Motivated by the relation (7 ), we introduce a transformation2 ) and get the conserved GP equation (4 ), of which the total particle number ${ \mathcal N }$ corresponding to 1SS is a constant,9 ), from (6 ) we get4 ). This is a space-time localized wave as ∣u∣2 decays exponentially in ∣x∣ at fixed t, and likewise, at fixed x, it decays exponentially in t. However, the total number of particles ${\int }_{-\infty }^{+\infty }| u{| }^{2}{\rm{d}}x$ is conserved.
$\begin{eqnarray}u=q\sqrt{\cosh (2\sqrt{\delta }t)},\end{eqnarray}$
by which we can remove the gain term from ( $\begin{eqnarray}{ \mathcal N }={\int }_{-\infty }^{\infty }| u{| }^{2}{\rm{d}}x=4| {a}_{1}| .\end{eqnarray}$
In fact, using the transformation ( $\begin{eqnarray}\begin{array}{rcl}| u{| }^{2} & = & 4{a}_{1}^{2}{\rm{sech}} (2\sqrt{\delta }t){{\rm{sech}} }^{2}\left[2{a}_{1}{\rm{sech}} (2\sqrt{\delta }t)\right.\\ & & \left.\times \left(x+\displaystyle \frac{2{b}_{1}}{\sqrt{\delta }}\sinh (2\sqrt{\delta }t)-{h}_{1}\cosh (2\sqrt{\delta }t)\right)\right],\end{array}\end{eqnarray}$
which provides a particle density (corresponding to one-soliton) governed by equation(When δ > 0, ∣u∣2 given in (11 ) is illustrated in figure 1, where we can see that the envelope becomes flatter when ∣t∣ increases but ∣u∣2 is indeed a conserved density.
Figure 1. Shape and motion of 1SS for equation ( |
When δ < 0, (11 ) reads12 ), and
$\begin{eqnarray}\begin{array}{rcl}| u{| }^{2} & = & 4{a}_{1}^{2}| \sec (2\sqrt{-\delta }t)| \,{{\rm{sech}} }^{2}\left[2{a}_{1}\sec 2(\sqrt{-\delta }t)\right.\\ & & \left.\times \left(x+\displaystyle \frac{2{b}_{1}}{\sqrt{-\delta }}\sin (2\sqrt{-\delta }t)-{h}_{1}\cos (2\sqrt{-\delta }t)\right)\right].\end{array}\end{eqnarray}$
By calculation, one can find that for given t (except periodic singularities), ∣u∣2 reaches its maximum $4{a}_{1}^{2}\sec (2\sqrt{-\delta }t)$ at $\begin{eqnarray*}x=-\displaystyle \frac{2{b}_{1}}{\sqrt{-\delta }}\sin (2\sqrt{-\delta }t)+{h}_{1}\cos (2\sqrt{-\delta }t).\end{eqnarray*}$
This curve can be viewed as the ‘top trace' of ( $\begin{eqnarray*}x^{\prime} (t)=-4{b}_{1}\cos (2\sqrt{-\delta }t)-2\sqrt{-\delta }{h}_{1}\sin (2\sqrt{-\delta }t\end{eqnarray*}$
as the velocity of the wave. The top trace is involved with trigonometric functions, which indicates oscillations of the wave, as depicted in figure 2. Note that there are periodic singularities (blowup points) at $\begin{eqnarray}({x}_{j},{t}_{j})=\left({\left(-1\right)}^{j+1}\displaystyle \frac{2{b}_{1}}{\sqrt{-\delta }},\,\displaystyle \frac{\pi }{2\sqrt{-\delta }}(\displaystyle \frac{1}{2}+j)\right),j\in {\mathbb{Z}}.\end{eqnarray}$
Here as a comment, we point out that the statement given in section 4.1 of Ref.[7] that ∣u∣2 takes finite amplitude at the above blowing-up points, is NOT correct.1
Figure 2. Shape and motion of 1SS for equation ( |
For the GP equation (4 ), it is easy to see from the equation that13 ). There are high-order conservation laws for equation (4 ). We will derive them in the next section.
$\begin{eqnarray*}| u{| }_{t}^{2}={({\rm{i}}{u}_{x}{u}^{* }-{\rm{i}}{{uu}}_{x}^{* })}_{x},\end{eqnarray*}$
which indicates that ∣u∣2 is a conserved density of fluid. Here * stands for complex conjugate. For the case δ < 0, one should consider the above conservation law at $t\in ({t}_{j},{t}_{j+1}),\,j\in {\mathbb{Z}}$, with tj given in (With regard to integrability, note that the GP equation (2 ) is connected to an integrable non-isospectral NLS equation14 ) belongs to the Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur (ZS-AKNS) hierarchy, with a Lax pair,4 ) is guaranteed if3 ).
$\begin{eqnarray}{\rm{i}}{Q}_{t}+{Q}_{{xx}}+2| Q{| }^{2}Q+2{\rm{i}}\alpha {({xQ})}_{x}=0\end{eqnarray}$
via a transformation $\begin{eqnarray}Q\,=\,q{{\rm{e}}}^{-\tfrac{{\rm{i}}\alpha }{2}{x}^{2}}.\end{eqnarray}$
Equation ( $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Phi }}}_{x} & = & \left(\begin{array}{ll}-{\rm{i}}\lambda & Q\\ R & {\rm{i}}\lambda \end{array}\right){\rm{\Phi }},\\ {{\rm{\Phi }}}_{t} & = & \left(\begin{array}{ll}-{\rm{i}}{QR}+2{\rm{i}}\lambda \alpha x-2{\rm{i}}{\lambda }^{2} & 2\lambda Q+{\rm{i}}{Q}_{x}-2\alpha {xQ}\\ 2\lambda R-{\rm{i}}{R}_{x}-2\alpha {xR} & {\rm{i}}{QR}-2{\rm{i}}\lambda \alpha x+2{\rm{i}}{\lambda }^{2}\end{array}\right){\rm{\Phi }},\end{array}\end{eqnarray*}$
where R = − Q*, spectral parameter λ is time-dependent, obeying λt = − 2αλ, and * stands for complex conjugate. Thus, integrability of the conserved GP equation ( $\begin{eqnarray*}Q=\sqrt{{\rm{sech}} (2\sqrt{\delta }t)}{{\rm{e}}}^{-\tfrac{{\rm{i}}\alpha }{2}{x}^{2}}u,\,\,R=-\sqrt{{\rm{sech}} (2\sqrt{\delta }t)}{{\rm{e}}}^{\tfrac{{\rm{i}}\alpha }{2}{x}^{2}}{u}^{* }\end{eqnarray*}$
and α(t) obeys (3. The conserved vector GP equation (5 )
3.1. Lax pair
The conserved vector GP equation (5 ) can be obtained from an integrable non-isospectral vector NLS equation16 )
$\begin{eqnarray}{\rm{i}}{{\boldsymbol{q}}}_{t}+{{\boldsymbol{q}}}_{{xx}}+2{({{\boldsymbol{q}}}^{\dagger }{\boldsymbol{q}}){\boldsymbol{q}}+2{\rm{i}}\alpha (x{\boldsymbol{q}})}_{x}={\boldsymbol{0}}\end{eqnarray}$
via the transformation $\begin{eqnarray}{\boldsymbol{u}}(x,t)=g{\left(t\right)}^{-\tfrac{1}{2}}{{\rm{e}}}^{\tfrac{{\rm{i}}\alpha }{2}{x}^{2}}{\boldsymbol{q}}(x,t).\end{eqnarray}$
Along a routine procedure in [15, 16], when the spectral parameter λ obeys λt = −2αλ, we get the Lax pair of equation ( $\begin{eqnarray}{\phi }_{x}=U\phi ,\,\,\quad U=-{\rm{i}}\lambda {{\rm{\Sigma }}}_{3}+{ \mathcal Q },\end{eqnarray}$
$\begin{eqnarray}{\phi }_{t}=V\phi ,\,\,\quad V = -2{\rm{i}}{\lambda }^{2}{{\rm{\Sigma }}}_{3}+2{\rm{i}}\lambda \alpha x{{\rm{\Sigma }}}_{3}+2\lambda { \mathcal Q } -2\alpha x{ \mathcal Q }-{\rm{i}}{{ \mathcal Q }}^{2}{{\rm{\Sigma }}}_{3}-{\rm{i}}{{ \mathcal Q }}_{x}{{\rm{\Sigma }}}_{3},\end{eqnarray}$
where $\begin{eqnarray}{{\rm{\Sigma }}}_{3}=\left(\begin{array}{ll}{I}_{n} & {\boldsymbol{0}}\\ {\boldsymbol{0}} & -1\end{array}\right),\,\,\quad { \mathcal Q }=\left(\begin{array}{ll}{\boldsymbol{0}} & {\boldsymbol{q}}\\ -{{\boldsymbol{q}}}^{\dagger } & 0\end{array}\right),\end{eqnarray}$
$\phi ={({\phi }_{1},{\phi }_{2},\cdots ,{\phi }_{n+1})}^{{\rm{T}}}$, ${\boldsymbol{q}}={({q}_{1},{q}_{2},\cdots ,{q}_{n})}^{{\rm{T}}}$ is n-component complex-valued vector field, In is the n × n identity matrix.3.2. Conservation laws
In the following, we construct infinitely many conservation laws for the conserved vector GP equation (5 ) and show that ${\sum }_{j=1}^{n}| {u}_{j}{| }^{2}$ is a conserved particle density. Infinite many conservation laws for the conserved vector GP equation (5 ) can be obtained either from the infinite conservation laws of the isospectral vector NLS equation (30 ) (see section 4 in [15]) coupled with the gauge transformation as listed in theorem 1 , or from those of the non-isospectral vector NLS equation (16 ) coupled with transformation (17 ). Here we will show the latter approach.
Start from the generalized ZS-AKNS spectral problem (18 ), i.e., in a scalar system,20 ) we get the Riccati equation set22 ) allows a series solution
$\begin{eqnarray}{\phi }_{j,x}=-{\rm{i}}\lambda {\phi }_{j}+{q}_{j}{\phi }_{n+1},\,\,\quad j=1,2,\cdots ,n,\end{eqnarray}$
$\begin{eqnarray}{\phi }_{n+1,x}=-\sum _{j=1}^{n}{q}_{j}^{* }{\phi }_{j}+{\rm{i}}\lambda {\phi }_{n+1}.\end{eqnarray}$
Introducing $\begin{eqnarray}{\omega }_{j}=\displaystyle \frac{{\phi }_{j}}{{\phi }_{n+1}},\quad j=1,2,\cdots ,n,\end{eqnarray}$
from ( $\begin{eqnarray}{\omega }_{j,x}=-2{\rm{i}}\lambda {\omega }_{j}+{q}_{j}+\sum _{i=1}^{n}{q}_{i}^{* }{\omega }_{i}{\omega }_{j},\quad j=1,2,\cdots ,n.\end{eqnarray}$
Noticing that the time evolution λt = − 2αλ of the spectral parameter λ implies $\lambda =a{\rm{sech}} (2\sqrt{\delta }t)$, where $a\in {\mathbb{C}}$, here we have chosen δ > 0 without loss of generality. The Riccati equation set ( $\begin{eqnarray}{\omega }_{j}=\sum _{k=1}^{\infty }{\omega }_{j,k}{\left(2{\rm{i}}a\right)}^{-k},\,\,\,j\,=\,1,2,\cdots ,n,\end{eqnarray}$
with $\begin{eqnarray}{\omega }_{j,1}=\cosh (2\sqrt{\delta }t){q}_{j},\end{eqnarray}$
$\begin{eqnarray}{\omega }_{j,k+1}=-\cosh (2\sqrt{\delta }t)\left({\omega }_{j,k,x}-\sum _{i=1}^{n}{q}_{i}^{* }\sum _{s=1}^{k-1}{\omega }_{i,s}{\omega }_{j,k-s}\right)\end{eqnarray}$
for j = 1, 2, ⋯ ,n.Next, from the last row of Lax pair (18a ) and (18b ), one can find18 ). The compatibility condition ${(\mathrm{ln}{\phi }_{n+1})}_{{xt}}={(\mathrm{ln}{\phi }_{n+1})}_{{tx}}$ leads to23 ) into the above equation as well as λt = − 2αλ and the expressions of C and D, by comparing terms of the same order in (2ia)−1, we get infinitely many conservation laws for equation (16 )5 ) can be obtained by substituting (17 ) into (26 ). We list the first two conserved densities and the associated fluxes (for the case δ > 0)
$\begin{eqnarray*}\begin{array}{rcl}{(\mathrm{ln}{\phi }_{n+1})}_{x} & = & -\displaystyle \sum _{j=1}^{n}{q}_{j}^{* }{\omega }_{j}+{\rm{i}}\lambda ,\\ {(\mathrm{ln}{\phi }_{n+1})}_{t} & = & \displaystyle \sum _{i=1}^{n}{C}_{i}{\omega }_{i}+D,\end{array}\end{eqnarray*}$
where Ci is the ith component of the vector C and D is the scalar function in ( $\begin{eqnarray}{\left({\rm{i}}\lambda -\sum _{i=1}^{n}{q}_{i}^{* }{\omega }_{i}\right)}_{t}={\left(\sum _{i=1}^{n}{C}_{i}{\omega }_{i}+D\right)}_{x}.\end{eqnarray}$
Then inserting the expansion ( $\begin{eqnarray}{({\rho }_{k}{({\boldsymbol{q}}))}_{t}=({J}_{k}({\boldsymbol{q}}))}_{x},\quad k=1,2,\cdots ,\end{eqnarray}$
where $\begin{eqnarray}{\rho }_{k}({\boldsymbol{q}})=-\sum _{j=1}^{n}{q}_{j}^{* }{\omega }_{j,k},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{J}_{k}({\boldsymbol{q}}) & = & \displaystyle \sum _{j=1}^{n}\left({\rm{i}}{\rm{sech}} (2\sqrt{\delta }t){q}_{j}^{* }{\omega }_{j,k+1}+2\alpha {{xq}}_{j}^{* }{\omega }_{j,k}\right.\\ & & \left.+{\rm{i}}{q}_{j,x}^{* }{\omega }_{j,k}\right),\quad k=1,2,\cdots \end{array}\end{eqnarray}$
are conserved densities and the associated fluxes, respectively. Infinitely many conservation laws of the conserved vector GP equation ( $\begin{eqnarray*}\begin{array}{rcl}{\rho }_{1}({\boldsymbol{u}}) & = & -\displaystyle \sum _{j=1}^{n}| {u}_{j}{| }^{2},\\ {\rho }_{2}({\boldsymbol{u}}) & = & \displaystyle \sum _{j=1}^{n}\cosh (2\sqrt{\delta }t)(-{\rm{i}}\alpha {{xu}}_{j}+{u}_{j,x}){u}_{j}^{* },\\ {J}_{1}({\boldsymbol{u}}) & = & {\rm{i}}\displaystyle \sum _{j=1}^{n}\left({u}_{j}{u}_{j,x}^{* }-{u}_{j}^{* }{u}_{j,x}\right),\\ {J}_{2}({\boldsymbol{u}}) & = & \cosh (2\sqrt{\delta }t)\sum _{j=1}^{n}\left(-{\rm{i}}| {u}_{j,x}{| }^{2}+\alpha x({u}_{j}^{* }{u}_{j,x}-{u}_{j}{u}_{j,x}^{* })\right.\\ & & \,\,\,\left.+\alpha | {u}_{j}{| }^{2}+{\rm{i}}{u}_{j}^{* }{u}_{j,{xx}}+{\rm{i}}{\rm{sech}} (2\sqrt{\delta }t)\sum _{i=1}^{n}| {u}_{i}{u}_{j}{| }^{2}\right).\end{array}\end{eqnarray*}$
The related conserved quantities are $\begin{eqnarray}{I}_{k}({\boldsymbol{u}})={\int }_{-\infty }^{+\infty }{\rho }_{k}({\boldsymbol{u}}){\rm{d}}x,\quad k=1,2,\cdots .\end{eqnarray}$
3.3. Gauge equivalence and NSS
In this subsection, we present NSS for the conserved vector GP equation (5 ) based on gauge transformation.
We find that both scalar and vector conserved GP equations (4 ) and (5 ) are gauge equivalent to the corresponding standard isospectral NLS equations. The connections are described in the following (where the scalar equation is a special case of the vector form).
The conserved vector GP equation (
$\begin{eqnarray}{\rm{i}}{{\boldsymbol{\psi }}}_{T}+{{\boldsymbol{\psi }}}_{{XX}}+2({{\boldsymbol{\psi }}}^{\dagger }{\boldsymbol{\psi }}){\boldsymbol{\psi }}={\boldsymbol{0}},\end{eqnarray}$
are connected via the transformation $\begin{eqnarray}\begin{array}{rcl}X(x,t) & = & g(t)x,\quad T(x,t)=\displaystyle \frac{1}{2\delta }\alpha (t),\\ {\boldsymbol{u}}(x,t) & = & {\boldsymbol{\psi }}(X,T)\sqrt{g(t)}{{\rm{e}}}^{\tfrac{{\rm{i}}\alpha }{2}{x}^{2}},\end{array}\end{eqnarray}$
where ${\boldsymbol{\psi }}{(X,T)=({\psi }_{1}(X,T),{\psi }_{2}(X,T),\cdots ,{\psi }_{n}(X,T))}^{{\rm{T}}}$.NSS of the vector NLS equation (30 ) is of the form [15]5 ) can be obtained by virtue of the gauge transformation (31 ).
$\begin{eqnarray}{\boldsymbol{\psi }}{\left(X,T\right)=2{\rm{i}}\sum _{j,k=1}^{N}{{\boldsymbol{\beta }}}_{j}{{\rm{e}}}^{{\theta }_{k}^{* }-{\theta }_{j}}\left({M}^{-1}\right)}_{{jk}},\end{eqnarray}$
where M(X, T) is an N × N matrix of entries $\begin{eqnarray}{M}_{{jk}}=\displaystyle \frac{{\beta }_{j}^{\dagger }{\beta }_{k}{{\rm{e}}}^{-({\theta }_{k}+{\theta }_{j}^{* })}+{{\rm{e}}}^{{\theta }_{k}+{\theta }_{j}^{* }}}{{\lambda }_{k}-{\lambda }_{j}^{* }},\end{eqnarray}$
${\beta }_{j}={({\beta }_{j,1},\cdots ,{\beta }_{j,n})}^{{\rm{T}}}$ is an n-component constant column vector, ${\theta }_{k}={\rm{i}}{\eta }_{k}X+2{\rm{i}}{\eta }_{k}^{2}T$. The above result can be presented in a more compact form $\begin{eqnarray}{\psi }_{l}(X,T)=2{\rm{i}}\displaystyle \frac{\det {M}_{l}(X,T)}{\det M(X,T)},\quad l\,=\,1,\cdots ,n,\end{eqnarray}$
where $\begin{eqnarray}{M}_{l}(X,T)=\left(\begin{array}{cccc} & & & {\theta }_{1}^{* }\\ & M(X,T) & & \vdots \\ & & & {\theta }_{N}^{* }\\ {\beta }_{1,l} & \cdots & {\beta }_{N,l} & 0\end{array}\right).\end{eqnarray}$
Then, the NSS of the conserved vector GP equation (3.4. Dynamics for the conserved vector GP equation (5 )
In this subsection, we investigate dynamics for the conserved vector GP equation (5 ).
3.4.1. Solutions for the case δ > 0
Taking N = 1 in the formula (34 ) and combined with the transformation (31 ), it is straightforward to obtain 1SS for the conserved vector GP equation (5 ). Let us consider the two-component case, i.e. fixing n = 2. The 1SS for equation (5 ) reads (cf [17])
$\begin{eqnarray}\begin{array}{rcl}\left[\begin{array}{l}{u}_{1}(x,t)\\ {u}_{2}(x,t)\end{array}\right] & = & {\beta }_{1}{{\rm{e}}}^{{\gamma }_{11}}(2{b}_{1}){\left({\rm{sech}} (2\sqrt{\delta }t)\right)}^{\tfrac{1}{2}}\\ & & \times {{\rm{e}}}^{{\rm{i}}{\zeta }_{1}}\,{\rm{sech}} ({\tau }_{1}+{\gamma }_{11}){{\rm{e}}}^{-\tfrac{{\rm{i}}\alpha }{2}{x}^{2}}.\end{array}\end{eqnarray}$
Here and after we take ${\eta }_{1}={a}_{1}+{\rm{i}}{b}_{1},{a}_{1},{b}_{1}\in {\mathbb{R}}$, denote $\begin{eqnarray}{\zeta }_{1}=2\left({a}_{1}x{\rm{sech}} (2\sqrt{\delta }t)+\displaystyle \frac{{b}_{1}^{2}-{a}_{1}^{2}}{\sqrt{\delta }}\tanh (2\sqrt{\delta }t)\right),\end{eqnarray}$
$\begin{eqnarray}{\tau }_{1}=2\left({b}_{1}x{\rm{sech}} (2\sqrt{\delta }t)-2\displaystyle \frac{{a}_{1}{b}_{1}}{\sqrt{\delta }}\tanh (2\sqrt{\delta }t)\right),\end{eqnarray}$
and also set ${\beta }_{1}^{\dagger }{\beta }_{1}={{\rm{e}}}^{-2{\gamma }_{11}}\gt 0$ to avoid singularities. Thus, we obtain the carrier wave $\begin{eqnarray}\begin{array}{rcl}\left[\begin{array}{c}| {u}_{1}(x,t){| }^{2}\\ | {u}_{2}(x,t){| }^{2}\end{array}\right] & = & \left[\begin{array}{c}| {\beta }_{\mathrm{1,1}}{| }^{2}\\ | {\beta }_{\mathrm{1,2}}{| }^{2}\end{array}\right]{{\rm{e}}}^{2{\gamma }_{11}}(4{b}_{1}^{2})\\ & & \times \ \left({\rm{sech}} (2\sqrt{\delta }t)\right)\,{{\rm{sech}} }^{2}({\tau }_{1}+{\gamma }_{11}).\end{array}\end{eqnarray}$
It can be verified easily that the total particle number is $\begin{eqnarray}{ \mathcal N }\equiv {\int }_{-\infty }^{\infty }(| {u}_{1}{| }^{2}+| {u}_{2}{| }^{2}){\rm{d}}x\,=\,4| {b}_{1}| ,\end{eqnarray}$
which is conserved.The carrier wave ∣ul(x, t)∣2, (l = 1, 2) provides a soliton traveling with a localized-like time-varying amplitude
$\begin{eqnarray}| {\beta }_{\mathrm{1,1}}{| }^{2}{{\rm{e}}}^{2{\gamma }_{11}}(4{b}_{1}^{2}){\rm{sech}} (2\sqrt{\delta }t),\end{eqnarray}$
top trajectory $\begin{eqnarray}x(t)=\displaystyle \frac{2{a}_{1}}{\sqrt{\delta }}\sinh (2\sqrt{\delta }t)-\displaystyle \frac{{\gamma }_{11}}{2{b}_{1}}\cosh (2\sqrt{\delta }t),\end{eqnarray}$
and velocity $\begin{eqnarray}{x}^{{\prime} }(t)=4{a}_{1}\cosh (2\sqrt{\delta }t)-\displaystyle \frac{{\gamma }_{11}}{{b}_{1}}\sqrt{\delta }\sinh (2\sqrt{\delta }t).\end{eqnarray}$
The above top trajectory can be further described according to the sign of $s=(4{a}_{1}{b}_{1}+{\gamma }_{11}\sqrt{\delta })(4{a}_{1}{b}_{1}-{\gamma }_{11}\sqrt{\delta })$: when s > 0, x(t) runs like $\sinh (2\sqrt{\delta }t)$, when s < 0, x(t) runs like $\cosh (2\sqrt{\delta }t)$, when s = 0, x(t) runs like ${{\rm{e}}}^{2\sqrt{\delta }t}$ or ${{\rm{e}}}^{-2\sqrt{\delta }t}$ and in particular x(t) is stationary when a1 = γ1,1 = 0, as shown in figures 3(a)–(d), respectively.
Figure 3. Shape and motion of 1SS given by ( |
3.4.2. Solutions for the case δ < 0
In the case of δ < 0, 1SS for equation (5 ) with δ < 0 can be written as5 ) with δ < 0 can be constructed in a similar way as mentioned in section 3.4 . As for dynamics, ∣u1(x, t)∣2 provides a wave traveling with amplitude $| {\beta }_{\mathrm{1,1}}{| }^{2}{{\rm{e}}}^{2{\gamma }_{11}}4{b}_{1}^{2}\left(\sec (2\sqrt{-\delta }t)\right)$, periodical top trajectory
$\begin{eqnarray}\begin{array}{rcl}\left[\begin{array}{l}{u}_{1}(x,t)\\ {u}_{2}(x,t)\end{array}\right] & = & {\beta }_{1}{{\rm{e}}}^{{\gamma }_{11}}(2{b}_{1}){\left(\sec (2\sqrt{-\delta }t)\right)}^{\tfrac{1}{2}}\\ & & {{\rm{e}}}^{{\rm{i}}{\tilde{\zeta }}_{1}}\,{\rm{sech}} ({\tilde{\tau }}_{1}+{\gamma }_{\mathrm{1,1}}){{\rm{e}}}^{-\tfrac{{\rm{i}}\alpha }{2}{x}^{2}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\tilde{\zeta }}_{1}=2\left({a}_{1}x\sec (2\sqrt{-\delta }t)+\displaystyle \frac{{b}_{1}^{2}-{a}_{1}^{2}}{\sqrt{-\delta }}\tan (2\sqrt{-\delta }t)\right),\end{eqnarray}$
$\begin{eqnarray}{\tilde{\tau }}_{1}=2\left({b}_{1}x\sec (2\sqrt{-\delta }t)-2\displaystyle \frac{{a}_{1}{b}_{1}}{\sqrt{-\delta }}\tan (2\sqrt{-\delta }t)\right).\end{eqnarray}$
Carrier wave of 1SS reads $\begin{eqnarray}\begin{array}{rcl}\left[\begin{array}{l}| {u}_{1}(x,t){| }^{2}\\ | {u}_{2}(x,t){| }^{2}\end{array}\right] & = & \left[\begin{array}{l}| {\beta }_{\mathrm{1,1}}{| }^{2}\\ | {\beta }_{\mathrm{1,2}}{| }^{2}\end{array}\right]{{\rm{e}}}^{2{\gamma }_{11}}(4{b}_{{1}^{2}})\\ & & \times \ \left(\sec (2\sqrt{-\delta }t)\right)\,{{\rm{sech}} }^{2}({\tilde{\tau }}_{1}+{\gamma }_{\mathrm{1,1}}).\end{array}\end{eqnarray}$
By calculating, it is found that the total particle number is $\begin{eqnarray}{ \mathcal N }\equiv {\int }_{-\infty }^{\infty }(| {u}_{1}{| }^{2}+| {u}_{2}{| }^{2}){\rm{d}}x\,=\,4| {b}_{1}| ,\end{eqnarray}$
which is conserved. Besides, infinitely many conserved quantities for equation ( $\begin{eqnarray}x(t)=\displaystyle \frac{2{a}_{1}}{\sqrt{-\delta }}\sin (2\sqrt{-\delta }t)-\displaystyle \frac{{\gamma }_{11}}{2{b}_{1}}\cos (2\sqrt{-\delta }t),\end{eqnarray}$
velocity $\begin{eqnarray}{x}^{{\prime} }(t)=4{a}_{1}\cos (2\sqrt{-\delta }t)-\displaystyle \frac{{\gamma }_{11}}{{b}_{1}}\sqrt{-\delta }\sin (2\sqrt{-\delta }t).\end{eqnarray}$
singularities appear at $({x}_{j}={(-1)}^{j}\tfrac{2{a}_{1}}{\sqrt{-\delta }},{t}_{j}=\tfrac{\pi }{2\sqrt{-\delta }}\left(\tfrac{1}{2}+j\right)),j\in {\mathbb{Z}}$. Dynamics are illustrated in figure 4(a), together with the density plot given in figure 4(b).
Figure 4. (a) 1SS of ∣u1(x, t)∣ given by ( |
4. Conclusions and remarks
In this paper, we have introduced a way to obtain a conserved GP equation with a parabolic potential. This idea was explained and illustrated in section 2 , where we started from the non-conserved GP equation (2 ) with a parabolic potential, by calculating its total particle number ${ \mathcal N }$ (see equation (7 )) associated with 1SS, we were led to the transformation (9 ), and the resulting GP equation (4 ) turns out to be conserved in terms of total particle number ${ \mathcal N }$. The conserved GP equation (4 ) contains a time-dependent coefficient g(t) to measure inter-particle interactions. This idea was then extended to the vector GP equation in section 3 and the conserved version is given in equation (5 ). The dynamics of some solutions are illustrated.
The idea of this paper might be applied to the differential-difference GP models, which will be investigated elsewhere.