Multi-symplectic integrator of the generalized KdV-type equation based on the variational principle

The variational principle is used to construct a multi-symplectic structure of the generalized KdV-type equation. Accordingly, the local energy conservation law, the local momentum conservation law, and the Cartan form of the generalized KdV-type equation are given. An explicit multi-symplectic scheme for the generalized KdV equation based on the Fourier pseudo-spectral method and the symplectic Euler scheme is constructed. Through a numerical examination, the explicit multi-symplectic Fourier pseudo-spectral scheme for the generalized KdV equation not only preserve the discrete global energy conservation law and the global momentum conservation law with high accuracy, but show long-time numerical stability as well.


Multi-symplectic integrator of the generalized KdV-type equation based on the variational principle
Yi Wei 1* , Xing-Qiu Zhang 1 , Zhu-Yan Shao 1 , Jian-Qiang Gao 1 & Xiao-feng Yang 2 the variational principle is used to construct a multi-symplectic structure of the generalized KdV-type equation. Accordingly, the local energy conservation law, the local momentum conservation law, and the cartan form of the generalized KdV-type equation are given. An explicit multi-symplectic scheme for the generalized KdV equation based on the fourier pseudo-spectral method and the symplectic euler scheme is constructed. through a numerical examination, the explicit multi-symplectic fourier pseudo-spectral scheme for the generalized KdV equation not only preserve the discrete global energy conservation law and the global momentum conservation law with high accuracy, but show long-time numerical stability as well.
In this paper, we aim to study the generalized KdV-type equation in the form The KdV equation is originally used to describe long waves propagating in a channel. The KdV equation can also describes the propagation of plasma waves in a dispersive medium 2,3 . The KdV equation and the mKdV equation are most popular soliton equations which have been extensively studied 2,4-11 , because these two equations possess many interesting properties of mechanism and geometry 12 . Based on the homogeneous balance of undetermined coefficients method (HBUCM) 13,14 , Yang et al. 13 proposed the definition and a multi-symplectic structure of the generalized KdV-type equation. Obviously, the KdV equation, the mKdV equation and the generalized KdV equation are special cases of the generalized KdV-type equation.
To understand the mechanism of complex physical phenomena, we should obtain solutions of the generalized KdV-type equation. There are some methods to obtain exact solutions, such as the first integral method 15 , the (G′/G)-expansion method 9 , the homogeneous balance method 16 , the modified simple equation method 17 , simplified Hirota's method 18 , and so on. However, the generalized KdV-type equation is a nonlinear partial differential equation (NLPDE), it is difficult to obtain the exact solutions or there is no exact solution. In these cases, it is natural to resort to the numerical methods 13 . A large amount of numerical methods have been applied on the KdV equation and the mKdV equation in the last few years, including quintic B-Spline basis functions 19 , finite difference scheme 20 , symplectic method 12 , multi-symplectic Preissmann box scheme 6 , multi-symplectic box schemes 21 , and so on.
Many physical properties of a system are closely related to the geometric structure of the equation 22 . This naturally requires that numerical methods can preserve exactly geometric structure during the simulation. Multi-symplectic method possesses the stability and effectiveness, and can preserve the multi-symplectic structure of the Hamiltonian system 14,23 . In the present paper, a multi-symplectic structure of the generalized KdV-type equation is given by the variational principle. To obtain a multi-symplectic structure of the generalized KdV-type equation based on variational principle, we need to cast Eq. (1.1) into a system of equations. Therefore, it will be helpful to give a detailed derivation of a multi-symplectic structure for the generalized KdV-type equation since this derivation will provide some guiding principles in finding multi-symplectic structures for other NLPDEs.
We also consider a multi-symplectic Fourier pseudo-spectral discretization for Eq. (1.2) and demonstrate its convergence by simulating the evolution of the soliton. The remainder of this paper is organized as follows. In section 2, a multi-symplectic structure of the generalized KdV-type equation is derived based on the variational principle. An explicit multi-symplectic Fourier pseudo-spectral scheme for the generalized KdV equation is given in section 3. In section 4, the solitary wave behaviors of the generalized KdV equation are simulated. In section 5, some conclusions are given.

Multi-Symplectic Structure of the Generalized Kdv-type equation Based on the Variational principle
In this section, a multi-symplectic structure of the generalized KdV-type equation is given by the variational principle. The covariant configuration space for Eq. (1.1) is denoted by X × U, where X = (x, t) represents the space of independent variables and U = (φ, u, v, ω, σ) represents the space of dependent variables. The internal variables φ, u, v, ω, and σ are defined to construct a multi-symplectic structure of the generalized KdV-type equation. The first order prolongation of X × U is defined to be U (1) represents the space consisting of the first order partial derivatives. Let ς: X → U be a smooth function and we suppose ς ∈ H 2 [X] × H 2 [X], where H 2 [X] is the second order Sobolev space defined on X. Then, its first prolongation is denoted by ( , , , , , , , , , , , , , , ) Corresponding to the Lagrangian density (2.2), the action functional is defined by . depend on the parameter β. Now, the variation of the action functional (2.1) is calculated as follows: If ξ, η, τ, φ, ϑ, ρ, and γ have compact support on M, then B = 0. In this case, with the requirement of δS = 0 and from Eq. (2.5), the variation ξ yields the local energy conservation law ) t x and the variation η yields the local momentum conservation law where E, F, I, and G are same as to Eq. (2.8).
For a conservative L, i.e., one that does not depend on x and t explicitly, Eqs (2.9) and (2.10) become the local energy conservation law and the local momentum conservation law, respectively 5 .
The variations τ, φ, ϑ, ρ, and γ yield the Euler-Lagrange equation If the condition that ξ, η, τ, σ, θ, ϑ, and ς having compact support on M is not imposed, then from the boundary integral B, the Cartan form can be defined as The multi-symplectic form of the generalized KdV-type equation is defined to be . d (2 14) L L Remark 1 Equation (2.11) is equivalent to a multi-symplectic structure of generalized KdV-type Eq. (1.1) as follows:   and Hamiltonian function . According to the multi-symplectic theory presented by  , the multi-symplectic conservation law in the wedge product form, the local energy conservation law and the local momentum conservation law for the generalized KdV Eq. (1.2) are as follows: and τ are spatial and temporal step lengths, the indexes i and j denote the discrete space and time dimensions. Denote u i j as the approximate value of u(x i , t j ). As we know, the first-order differential operator ∂ x yields the Fourier spectral differentiation matrix D. Here, D is an N×N anti-symmetry matrix with elements (N is an even number) Using the notations The variational equation associated with Eq. (3.5) is Taking the wedge product with z d i on both sides of Eq. (3.6) and noticing where δ + t and δ − t are the forward and backward difference operators, respectively: T an efficient stable explicit scheme for the generalized KdV Eq. (1.2) is obtained as follows: Eliminating the auxiliary variables and re-writing the equations, a compact form is obtained as follows: www.nature.com/scientificreports www.nature.com/scientificreports/ Theorem 2. The discrete scheme (3.13) has N full-discrete multi-symplectic conservation laws  [ 20,40] L R with the periodic boundary condition ( 3 22) and the initial condition We fix the space step = .
h 0 2 and the time step τ = × − 1 10 4 for the scheme (3.14). Based on the multi-symplectic theory of   The errors of the discrete global energy conservation law Error E and the global momentum conservation law Error I on the jth time level are defined as follows: where E 0 and I 0 are the initial value of the discrete global energy and the global momentum. Now we define the error as follows: Applying the scheme (3.14) to simulate the generalized KdV Eq. (3.20) with the periodic boundary condition (3.22) and the initial conditions (3.23) up to t = 20, three-dimensional waveform figure of numerical solutions (Fig. 1a (t = 1) and Fig. 1b (t = 20)), two-dimensional waveform figure of numerical solutions (Fig. 2a (t = 1) and Fig. 2b (t = 20)), three-dimensional error figure (Fig. 3a), two-dimensional error figure (Fig. 3b), the global energy error figure (Fig. 4a) and the global momentum error figure (Fig. 4b) are obtained as follows: Figure 1 shows that waveform of numerical solutions does not change with time by applying the multi-symplectic Fourier pseudo-spectral method to Eq. (3.20). It indicates that the basic geometric properties of Eq. (3.20) can be well maintained by the numerical soliton solutions. From Fig. 2a,b, u decreases gradually and tends to zero with → ∞ x , which is consistent with the exact solution (3.21), and the numerical solutions nearly overlap the exact solution, which implies high accuracy of the multi-symplectic Fourier pseudo-spectral scheme. From the Fig. 3a,b, it is shown that the errors between the exact solution and the numerical solution are up to 10 −9 , and the errors are mainly from the boundary conditions. From Fig. 4a,b, the multi-symplectic Fourier pseudo-spectral method to the generalized KdV equation can well maintain two important geometric properties  www.nature.com/scientificreports www.nature.com/scientificreports/ of the system, which are the global energy conservation law and the global momentum conservation law. Obviously, when we take 10 −5 time steps, the multi-symplectic Fourier pseudo-spectral method to the generalized KdV equation can preserve the discrete global energy conservation law and global momentum conservation law quite well, which implies long-time numerical stability of the multi-symplectic pseudo-spectral method.

conclusions
The generalized KdV-type equation, which can degenerate to the mKdV equation and the generalized KdV equation, is given. The variational principle is successfully used to establish a multi-symplectic structure for the KdV-type equation. Based on the variational principle, we also obtain the multi-symplectic structure, local energy conservation law and local momentum conservation law of the generalized KdV-type equation, which are identical to the results by using the HBUCM 6,13 . An explicit multi-symplectic scheme for the generalized KdV equation based on the Fourier pseudo-spectral method and the symplectic Euler scheme are constructed. Through a numerical examination, the explicit multi-symplectic Fourier pseudo-spectral scheme for the generalized KdV equation not only preserve the discrete global energy conservation law and the global momentum conservation law with high accuracy, but show long-time numerical stability as well.
The performance of variational principle is found to be simple and efficient. Moreover, similar to the process of the generalized KdV-type equation, multi-symplectic structures of some NLPDEs can be obtained.