High-order Runge–Kutta structure-preserving methods for the coupled nonlinear Schrödinger–KdV equations

Abstract

A novel class of high-order Runge–Kutta structure-preserving methods for the coupled nonlinear Schrödinger–KdV equations is proposed and analyzed. With the aid of the quadratic auxiliary variable, an equivalent system is obtained from the original problem. The Fourier pseudo-spectral method is employed in spatial discretization and the symplectic Runge–Kutta method is utilized for the resulting semi-discrete system to arrive at a high-order fully discrete scheme. Simultaneously, the conservation of the original multiple invariants for the schemes are rigorously proven. Numerical experiments are performed to verify the theoretical analysis.

Introduction

The coupled nonlinear Schrödinger–KdV (NLS–KdV) equations [4] have been widely used to describe the nonlinear dynamics of one-dimensional Langmuir and ion-acoustic waves in a system of coordinates moving at the ion-acoustic speed, namely,

$$\left\{\begin{array}{cc}\hfill & \mathrm{i}\varepsilon u_t+p{u}_{xx}-qvu-s{\left|u\right|}^{2}u=0,\hfill \\ \hfill & v_t+\alpha v_{xxx}+{\left(\beta v^m+\rho {\left|u\right|}^2\right)}_x=0,x\in \Omega ,t>0,\hfill \end{array}\right.$$

with the periodic boundary conditions and the initial conditions

$$u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),x\in \Omega .$$

Here $\varepsilon >0$ is a positive constant, $\Omega \subseteq \mathbb{R}$ is a bounded domain, the complex-valued function $u=u(x,t)$ and the real-valued function $v=v(x,t)$ describe electric field of Langmuir oscillations and low-frequency density perturbation, respectively.

As is known, the NLS–KdV equations (1.1) conserve the following energy of oscillations

$$\mathcal{E}\left(t\right)={\int }_{\Omega }\left(\frac{q\beta }{m+1}{v}^{m+1}+p\rho {\left|u_x\right|}^2+q\rho v{\left|u\right|}^2+\frac{s\rho }{2}{\left|u\right|}^4-\frac{q\alpha }{2}{\left(v_x\right)}^2\right)dx\equiv \mathcal{E}\left(0\right),t\ge 0,$$

the number of plasmons

$${\mathcal{P}}_1\left(t\right)={\int }_{\Omega }{\left|u\right|}^{2}dx\equiv {\mathcal{P}}_1\left(0\right),t\ge 0,$$

the number of particles

$${\mathcal{P}}_2\left(t\right)={\int }_{\Omega }vdx\equiv {\mathcal{P}}_2\left(0\right),t\ge 0,$$

and the momentum

$$\mathcal{M}\left(t\right)={\int }_{\Omega }\left(q{v}^2-2\rho \varepsilon \mathrm{Im}\left(u{\overline{u}}_x\right)\right)dx\equiv \mathcal{M}\left(0\right),t\ge 0.$$

Extensive numerical studies have been carried out for the NLS–KdV equations (1.1) in the literatures, such as the decomposition method [22], Haar wavelet collocation method [30], homotopy perturbation method [2], [23], meshless method [18], variational iteration method [1]. However, these mentioned schemes cannot conserve the invariants (1.3)–(1.6). It is clear to see that the solution of energy-conserving schemes cannot produce nonlinear blow-up for the numerical simulation in [33], [42]. Therefore, there are some literatures concerning the structure-preserving schemes for the NLS–KdV equations (1.1). Bai et al. [5], [6] proposed the finite element methods for the NLS–KdV equations. Zhang et al. [43], [44] constructed two structure-preserving schemes for the NLS–KdV equations. Cai et al. [12] developed the temporal second- and fourth-order schemes and the schemes are decoupled as well as exactly conserve three invariants (1.3)–(1.5) simultaneously. Xie and Yi [39] proposed a linearly-implicit conservative compact finite difference method for the NLS–KdV equations (1.1), which conserve the invariants (1.3)–(1.5). Unfortunately, a majority of those energy-preserving schemes are only second-order accurate in time. As well known, when the large time step is chosen, compared with the second-order accurate scheme, the high-order structure-preserving schemes have better robustness and much smaller numerical error [15], [21]. Thus, it is interesting to devise high-order accurate multiple invariants conserving methods for solving the NLS–KdV equations (1.1), which are able to preserve the invariants in the discrete solution.

In recent years, a lot of high-order structure-preserving schemes have been successfully applied for Hamiltonian systems, including energy-preserving continuous stage Runge–Kutta (RK) methods [24], [29], [37], high-order averaged vector field (AVF) methods [27], [31], Hamiltonian Boundary Value Methods (HBVMs) [7], [9], [10] etc. Overall, there exist few works devoted to the development of high-order structure-preserving methods that preserve multiple invariants [8], [11], [16], [26], [28]. In this paper, we focus on constructing a class of high-order multiple invariants conserving methods for the NLS–KdV equations (1.1), which are called quadratic auxiliary variable (QAV) approach, motivated by the ideas of high-order accurate energy-preserving methods proposed in [13], [38].

This paper is organized as follows. In Section 2, we first reformulate the original NLS–KdV equations (1.1) into an equivalent form with a modified quadratic energy, and the Fourier pseudo-spectral method is employed in spatial discretization, and then a class of high-order multiple invariants conserving schemes based on the symplectic RK methods for extending to the fully-discrete high-order energy-preserving schemes. A fixed point iterative scheme for the proposed method is presented in Section 3. In Section 4, numerical experiments for the NLS–KdV equations (1.1) are carried out to illustrate the capability and accuracy of the method. Finally, we give some conclusions.