Numerical simulation of a generalized nonlinear derivative Schrödinger equation

1.   Introduction

In this paper, we consider a generalized nonlinear derivative Schrödinger (GNLDS) equation ^[1,2,3] as follows

with periodic boundary condition

and initial condition

where $u = u(x, t)$ is a complex-valued function, $\gamma_1$ denotes the strength of higher-order linear and nonlinear effects, and $\overline{u}$ means taking the conjugate of $u$.

The GNLDS equation is a integrable system which can be viewed as a particular case of a spin system derived in ^[1], which advanced the fourth order generalized nonlinear Schrödinger equation for the isotropic Heisenberg ferromagnetic spin chain in the next order of lattice parameter through identifying the underlying geometry of the system, and demonstrated that the spin system with the lattice parameter $\beta = -\frac{5}{2}v$ ($v$ is equivalent to $\gamma_1$ in this paper) is equivalent to the higher order integrable generalized nonlinear Schrödinger evolution equation, that is, (1.1). In 2013, Wang, Porsezian and He ^[2] obtained the first-order rogue wave solution and higher-order rogue waves from the GNLDS evolution equation by using Darboux transformation and Taylor expansion, moreover the compressed effects of parameter $\gamma_1$ was talked over. Zhang and Chen ^[3] studied the robust inverse scattering transformation transformation to construct the high-order rogue wave based on the GNLDS equation in 2019. For the rogue waves and solitary waves, we refer to ^[4,5,6] and references therein.

The famous mathematical model nonlinear Schrödinger equation ^[7,8] has been widely used in the study of quantum physics and theoretical chemistry, so various highly accurate and effective numerical methods are required and have been developed for the NLS equation, for instance [9–16], finite difference method [17–23], finite element method ^[24,25,26], discontinuous Galerkin method ^[27,28], meshless method ^[29,30], and Runge-Kutta or Crank-Nicolson pseudo-spectral method ^[31,32,33], time-splitting pseudo-spectral method ^{[34,35,36,37,38]}, the spectral and pseudo-spectral method ^[39,40,41], split step spectral method ^[42], structure-preserving algorithms^[43,44,45], and other methods ^{[46,47,48,49,50,51]}. These different numerical methods have their own advantages and disadvantages. For the comparison between them, we refer to ^[52,53,54] and references therein. Although above numerical methods have been researched for NLS, as far as we know, there are no corresponding literatures which focused on aspects of numerical simulation of (1.1). Thus, to construct respectively numerical methods is our main work in this paper.

After simple observation and analysis of the equation, we find that

so we simplify the Eq (1.1), then the original question becomes

Lemma 1.1. The initial-boundary value problem (1.5)–(1.7) preserves the total mass

and energy

Remark 1.1. ${\mathrm{Q}}(t)$ and ${\mathrm{E}}(t)$ are the so-called mass (or charge) and energy, the process of certification is in ${\mathrm{Appendix}}$ ${\mathrm{A}}$.

Hence, to construct an accurate and stable numerical method preserving the total mass and energy is an interesting and challenging issue. However, it is difficult and challenging for us to construct a finite difference scheme which preserves the total mass and energy because of the higher order linear terms and some higher order nonlinear terms with derivatives. For this reason, we first propose two nonlinear two-level schemes, one is mass-preserving and the other is energy-preserving. In further study, we finally construct a three-level mass-preserving and energy-preserving scheme, the three schemes will be shown in the second part. Therefore, it is an exciting problem to explore what will happen to simulate the first-order rogue wave solution and other physical phenomena through our numerical simulation. It is our prime purpose to reply this question in this paper.

The remaining sections of this paper are arranged as follows. Finite difference schemes are constructed and the corresponding discrete conservation laws are proposed in Section 2; Section 3 is devoted to showing the conclusion of our numerical experiments which illustrate the effectiveness of the proposed methods; Finally, Section 4 contains the conclusion of this paper and the researches that need to be progressed in the future.

2.   Finite difference schemes and their conservative properties

In order to solve the problem (1.5)–(1.7) with the difference scheme, the solution area $\Omega = \{(x, t)|a\leq x\leq b, 0\leq t\leq {\mathrm{T}}\}$ will be divided. Take two positive integers $N$ and $J$, let $h = \frac{(b-a)}{J}, \tau = \frac{T}{N}$, and denote $\Omega_{h} = \{x_{j} = a+jh|j = 0, 1, ...J\}$, $\Omega_{\tau} = \{t_{n} = n\tau|n = 0, 1, ...N\}$.

Assuming that $\omega = \{\omega_{j}^{n}|(x_{j}, t_{n})\in \Omega_{h}\times\Omega_{\tau}\}$ is a grid function. We introduce the following standard difference operators:

We denote by $\mathbb V_h = \{v|v = (v_0, v_1, ..., v_J), v_0 = v_J\}$ a space of complex-valued grid functions defined on $\Omega_h$, we always use $v_{-1}^n = v_{J-1}^n$ and $v_{1}^n = v_{J+1}^n$ if they are involved. For any complex-valued grid functions $u, v\in\mathbb V_h$, we denote the discrete inner product as $(u, v) = h\sum\limits_{j = 0}^{J-1}u_j\overline{v}_j$, where $\overline{v}_j$ is the conjugate of $v_j$. The discrete $L_p$ norm $||\cdot||_p$, the $H^1$ semi-norm $|\cdot|_1$, the $H^2$ semi-norm $|\cdot|_2$ and the maximum norm $||\cdot||_{\infty}$ over $\mathbb V_h$ are defined by

respectively. For simplicity, we use $||v||$ to denote $||v||_2$.

2.1.   The three Crank-Nicolson-type finite difference schemes

In this section, we present three Crank-Nicolson-type finite difference schemes. Two of them are the two-level mass-preserving scheme or the two-level energy-preserving scheme and the third is a three-level mass-preserving and energy-preserving scheme. We denote the three schemes by CNFD-A, CNFD-B and CNFD-C as follows:

2.2.   Conservative properties

Corresponding to the conservation laws (1.8) and (1.9) preserved by the continuous problem (1.5)–(1.7), the scheme CNFD-C conserves the similar total mass and energy in the discrete level, the scheme CNFD-B conserves the corresponding total energy in the discrete level, the scheme CNFD-A conserves the corresponding total mass in the discrete level.

Theorem 2.1. Suppose $\{u_j^n|0\leq j\leq J, 0\leq n\leq N\}$ is the solution of CNFD-A scheme, then CNFD-A scheme has a conservation law of the following form with respect to discrete mass:

where

Theorem 2.2. Suppose $\{u_j^n|0\leq j\leq J, 0\leq n\leq N\}$ is the solution of CNFD-B scheme, then CNFD-B scheme has a conservation law of the following form with respect to discrete energy:

where

Theorem 2.3. Suppose $\{u_j^n|0\leq j\leq J, 0\leq n\leq N\}$ is the solution of CNFD-C scheme, then CNFD-C scheme has conservation laws of the following forms with respect to discrete mass and discrete energy:

where

Remark 2.1. $Q^n$ and $E^n$ are the so-called discrete mass (or charge) and energy respectively. The proofs of the three theorems are given in ${\mathrm{Appendix}}$ ${\mathrm{B}}$.

3.   Numerical examples

In this section, we construct several typical numerical examples to verify conservation laws and convergence rates, and simulate some phenomena of the solitary-wave collisions and the first-order rogue wave solution described by GNLDS equation.

Example 3.1. Mass and energy conservations laws: We give several numerical results out to validate the mass and energy conservations of the three schemes given in Theorems 2.1–2.3. Specifically, we take the initial condition and boundary condition as $\varphi(x) = {\mathrm{sech(x)exp(2ix)}}$, the periodic boundary condition $\frac{\partial^{m}u}{\partial{x^{m}}}(a, t) = \frac{\partial^{m}u}{\partial{x^{m}}}(b, t)$, $m = 0, 1, 2, 3$, and we consider the case $\gamma_1 = 0.5$. Figures 1–4 reveal the development of the mass and energy values of the corresponding numerical solutions for the three schemes. We can observe that, for CNFD-A scheme and CNFD-B scheme

for CNFD-C scheme,

which illustrates that the CNFD-A scheme preserves total mass conservation, the CNFD-B scheme preserves total energy conservation, and the CNFD-C scheme preserves both total energy and mass conservation in discrete sense.

Example 3.2. Convergence rates: In this example, we take the initial condition and the periodic boundary condition as $\varphi(x) = {\mathrm{sech(x)exp(2ix)}}$, $\frac{\partial^{m}u}{\partial{x^{m}}}(a, t) = \frac{\partial^{m}u}{\partial{x^{m}}}(b, t)$, $m = 0, 1, 2, 3$ to verify the convergence rates. We consider the case $\gamma_1 = 0.5$, then we compute the $L_{\infty}$ and $L_2$ error norms of the solution at $T = 1$ and use the same spacing h in each spatial direction. For convenience, we denote the $L_{\infty}$ and $L_2$ error norms as

and assume $E_{\infty}(h, \tau) = O(\tau^p+h^q)$, then if $\tau$ is small enough, $E_{\infty}(h, \tau) = O(\tau^p+h^q)\approx O(h^q)$, so $E_{\infty}(h_1, \tau)/E_{\infty}(h_2, \tau)\approx (h_1/h_2)^q$, and hence $q\approx \frac{{\mathrm{ln}}(E_{\infty}(h_1, \tau)/E_{\infty}(h_2, \tau))}{{\mathrm{ln}}(h_1/h_2)}$ is the convergence order with respect to the spatial step size. Likewise, if $h$ is small enough, $p\approx \frac{{\mathrm{ln}}(E_{\infty}(h, \tau_1)/E_{\infty}(h, \tau_2))}{{\mathrm{ln}}(\tau_1/\tau_2)}$ is the convergence order with respect to the temporal step size. The algorithm of $L_2$ error norm is consistent with $L_{\infty}$ error norm.

Firstly, we measure the temporal errors and convergence orders. Fix the spatial step size $h = 0.005$ sufficiently small such that the spatial error is negligible as compared with the temporal error. Tables 1, 3 and 5 show that the convergence order of the three schemes with respect to temporal variable are all about 2 when $h = 0.005$, $\tau = 1/70$, $\tau = 1/80$, $\tau = 1/90$, and $\tau = 1/100$.

Through varying $h$ and fixing $\tau = 1/8000$ small enough to avoid contamination of the temporal, we further measure the spatial errors and convergence orders. It can be seen from Tables 2, 4 and 6 show that the convergence order of three schemes are all about 2 with respect to the spatial step size.

Example 3.3. The influence of $\gamma_1$ on the motion of two solitary waves: In this section, we apply CNFD-C scheme to research the interaction between two solitary waves through changing the value of $\gamma_1$. The initial condition is chosen as

which illustrates two solitary waves are initially located at $x = 10$ and $x = -10$ and the periodic boundary condition $\frac{\partial^{m}u}{\partial{x^{m}}}(a, t) = \frac{\partial^{m}u}{\partial{x^{m}}}(b, t)$, $m = 0, 1, 2, 3$. We consider the following cases:

Figures 5 and 6 display the interaction of two solitary waves under different values of $\gamma_1$ for GNLDS equation under cases (1)–(4). It can be vividly seen that two initially well-separated solitons move towards each only on the cases $\gamma_1 > 0$, with the motion of the waves, they will produce a peak when they collide, after separation, they will continue to move in the original direction. When $\gamma_1\leq 0$, compared with $\gamma_1 > 0$, the two solitons move in the opposite direction at the beginning and there are no collisions. In addition, we find that the velocity of the waves is proportional to the absolute value of $\gamma_1$.

Example 3.4. The first-order rogue wave: We apply the numerical solution obtained by the CNFD-C scheme to simulate the first-order rogue wave (the limit of the breather solution) ^[2] in $\{(x, t)\in (a, b)\times(0, T)|-8\leq x\leq 8, -4\leq t\leq 4\}$. And we also increase the value of $\gamma_1$ to observe and compare the dynamical evolution of the analytic solutions and the numerical solutions. We take the initial value of the analytic solution by

take the periodic boundary values of the analytic solution by

and consider the following cases:

We show Figures 7–16 as follows, it can be seen that the numerical solutions agree well with the analytic solutions and they have approximately equal amplitudes under cases (1)–(5) which illustrates the effectiveness of CNFD-C.

4.   Conclusions and further questions

In conclusion, we have proposed three different finite difference schemes for ${\mathrm{GNLDS}}$ equation and we proved the conservation properties of the equation with periodic boundary condition. Moreover, the corresponding discrete conservation laws of the three schemes were advanced in discrete level. Among them, obviously CNFD-C scheme had better conservation properties. Furthermore, we gave several typical numerical examples to confirm the corresponding convergence rates and conservation laws for the three schemes in discrete sense. In examples 3 and 4, we used CNFD-C scheme to simulate the collision of solitary waves and the first-order rogue wave solution. These numerical experiments verified the effectiveness of the proposed schemes.

However, the theoretical analysis of the schemes is a greater difficulty and challenge for us, which will be our further exploration direction in the future research process. We will also consider perfecting some work on the time fractional order of the generalized nonlinear Schrödinger equation, which is also a challenge for us in the future. In addition, it can be seen from the energy expression that the coefficient in front of the $H^1$ semi-norm of the solution of the original equation is negative, then in the case of energy conservation, the $H^1$ semi-norm cannot be controlled, and it may tend to infinity and the blow-up phenomenon may occur, and this article has not been optimized for this situation. Therefore, in the following work, readers can specifically optimize a format for blow-up phenomena.

Appendix A: The proof of conservation laws for problem (1.5)–(1.7).

Here, we give a brief proof of Lemma 1.1.

Proof. Make inner product of both sides of (1.5) with $u(x, t)$ at the same time, that is, multiply both sides of the equation by $\overline{u}(x, t)$, and integrate x from $a$ to $b$. Then take the imaginary part of it, we get

${\mathrm{Re}}(v)$ and ${\mathrm{Im}}(v)$ mean taking the real part and imaginary part of $v$ respectively, we can get $\frac{d}{dt}Q(t) = 0$, where

Then (1.8) holds. Likewise, make inner product of both sides of (1.5) with $u_t(x, t)$ at the same time, and then take the real part of both sides of (1.5), we can get

Due to the periodic boundary conditions, we get

the last five items of (5.3) are calculated similarly, so we obtain

we can get $\frac{d}{dt}E(t) = 0$, where

This implies (1.9) holds. Therefore we complete the proof of Lemma 1.1.

Appendix B: The proof of conservation laws for three schemes in discrete sense.

Here, we give the proof of Theorem 2.3 because the proof processes of Theorems 2.1–2.2 are similar to Theorem 2.3, so we only give the proof ideas for Theorems 2.1–2.2 and omit the processes of them.

Proof. Make the inner product of (2.7) and $(u^{n+1}+u^{n-1})$ and take the imaginary part of the equation to get

we can get $Q^n = Q^{n-1} = \cdot\cdot\cdot = Q^0$, where

Next, make the discrete inner product of (2.7) and $\delta_{\hat t}u^n$, and then take the real part, we obtain

we can get $E^n = E^{n-1} = \cdot\cdot\cdot = E^0$, where

Therefore Theorem 2.3 holds.

Proof. We make the discrete inner product of (2.1) and $u^{n+\frac{1}{2}}$, and then take the imaginary part, therefore Theorem 2.1 holds.

Proof. We make the discrete inner product of (2.4) and $\delta_{t}^{+}u^n$, and then take the real part, therefore Theorem 2.2 holds.

Conflict of interest

There are no conflicts of interest to declare.