Numerical calculation of N-periodic wave solutions to coupled KdV–Toda-type equations

Abstract

In this paper, we study the N-periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N-periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N-periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

1. Introduction

Nonlinear evolution equations (both continuous and discrete) describe a variety of physical systems, at different scales from elementary particle models, to atomic and molecular physics, etc. The most useful nonlinear systems are of course the integrable ones. The progress in the theory of solitons and integrable systems has allowed the study of many nonlinear problems in mathematics and physics. Development in this field enables one to find remarkable solutions (say, soliton solutions and periodic wave solutions) of nonlinear equations which appear in various fields of physics. Those solutions play significant roles in understanding fundamental properties of physical systems.

In this paper, we focus on numerical calculation of the N-periodic wave solutions to coupled Korteweg–de Vries (KdV)–Toda-type soliton equations. The N-periodic wave solutions mentioned here represent periodic analogues of soliton solutions, and in general case N-periodic wave solution is a periodic generalization of N-soliton solution or multiple collision of N solitons [1]. Sometimes, this kind of solutions are also called algebro-geometric solutions and finite-genus solutions. It is known that some periodic solutions tend to the soliton solutions under a small amplitude limit.

Much work has already been done on periodic wave solutions. The pioneering work was made by Novikov and Dubrovin [2–4], Lax [5], Its & Matveev [6], McKean & Moerbeke [7] in the 1970s. After that, some classical methods, such as the inverse scattering method [8–10], the algebro-geometric approach [11–17] and the direct method [1,18–22], were applied to solve periodic waves. However, comparing with soliton waves, the periodic wave solutions are more complicated and it is difficult to give some detailed explicit expressions. Therefore, many researchers turn to numerical calculations. Recent work includes the numerical approach via the Riemann–Hilbert problem [23,24] and the spectral method [25–27]. Here we will calculate the periodic waves numerically based on the direct method [1,18,19,28].

In [1,18], Nakamura first proposed the conditions for having N-periodic waves to nonlinear evolution equations which can be reduced to some certain type of bilinear equations, such as KdV, mKdV, nonlinear Schrödinger and some other equations. And then in [19], Hirota suggested researchers investigate whether the soliton equations written in bilinear form exhibit 3-periodic wave solutions or not by this condition. Recently, motivated by this, we presented a numerical process to calculate the 3-periodic waves to some KdV-type and Toda-type equations and the mKdV–sine-Gordon equation [29–31].

In this paper, we will apply the conditions to the coupled KdV–Toda-type equations and give a numerical procedure to calculate their N-periodic wave solutions. Here ‘coupled KdV–Toda-type’ means that, with some suitable variable transformations and auxiliary variables, a nonlinear equation can be transformed into the following bilinear form:

$$F_1(D_t,D_z,D_x,\sinh ({\delta }_1{D}_m),\sinh ({\delta }_2{D}_n),\dots ,c_1)\hspace{0.17em}f\cdot f=0$$ 1.1

and

$$F_2(D_t,D_z,D_x,\sinh ({\delta }_1{D}_m),\sinh ({\delta }_2{D}_n),\dots ,c_2)\hspace{0.17em}f\cdot f=0,$$ 1.2

where F₁, F₂ are even functions of D_t, D_z, D_x, D_m, D_n, …, c₁, c₂ are integral constants, and the D operator [28] is defined by

$$D_t^m{D}_x^{k}a(t,x)\cdot b(t,x)=\frac{{\mathrm{\partial }}^m}{\mathrm{\partial }{s}^m}\frac{{\mathrm{\partial }}^k}{\mathrm{\partial }{y}^k}a(t+s,x+y)b(t-s,x-y){|}_{s=0,y=0},m,k=0,1,2,\dots ,$$ 1.3

and

$$\exp (\delta D_n)a(n)\cdot b(n)=a(n+\delta )b(n-\delta ).$$ 1.4

Many soliton equations can be viewed as coupled KdV–Toda-type equations. Here we will study seven of them, including the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak (BM) lattice, the semi-discrete KdV equation, the (2 + 1) dimensional Leznov lattice and a relativistic Toda (rToda) lattice. To give a direct impression of the coupled KdV–Toda-type equation, we show two of them in the following. One is the coupled Ramani equation [32]

$$u_{xxxxxx}+15u_{xx}{u}_{xxx}+15u_x{u}_{xxxx}+45u_x^2{u}_{xx}-5(u_{xxxt}+3u_{xx}{u}_t+3u_x{u}_{xt})-5u_{tt}+18v_x=0$$ 1.5

and

$$v_t-v_{xxx}-3v_x{u}_x-3v{u}_{xx}=0,$$ 1.6

which can be transformed into the bilinear form

$$(D_x^6-5D_x^3{D}_t-5D_t^2+9D_x{D}_z+c_1)f\cdot f=0$$ 1.7

and

$$(D_z{D}_t-D_z{D}_x^3-6v_0{D}_x^2+c_2)f\cdot f=0,$$ 1.8

by the dependent variable transformation

$$u=u_0+{(\ln f)}_x\text{and}v=v_0+{(\ln f)}_{xz},$$ 1.9

where z is an auxiliary variable and c₁, c₂ are integral constants. The other is the BM lattice [33]

$$\begin{array}{rl}\hspace{0.17em}& \frac{\text{d}}{\text{d}t}{a}_n=c_{n+1}-c_{n-1},\end{array}$$ 1.10

$$\begin{array}{rl}\hspace{0.17em}& \frac{\text{d}}{\text{d}t}{b}_n=a_{n-1}{c}_{n-1}-a_n{c}_n\end{array}$$ 1.11

$$\begin{array}{r}\text{and}\frac{\text{d}}{\text{d}t}{c}_n=c_n(b_n-b_{n+1}),\end{array}$$ 1.12

which can be transformed into the bilinear form

$$[D_z{D}_t-4{\sinh }^2(D_n)+c_1-2]f_n\cdot f_n=0$$ 1.13

and

$$[D_t^2-2D_z\sinh (D_n)+c_2\cosh (D_n)]f_n\cdot f_n=0,$$ 1.14

by the dependent variable transformation

$$a_n=\frac{D_t^2{f}_n\cdot f_n}{2f_{n+1}{f}_{n-1}},b_n=\frac{D_t{f}_{n-1}\cdot f_n}{f_{n-1}{f}_n}\text{and}{c}_n=\frac{f_{n+1}{f}_{n-1}}{f_n^2},$$ 1.15

where z is an auxiliary variable and c₁, c₂ are integral constants.

In the case of single KdV-type bilinear equations, as shown by Nakamura [1] and Hirota & Ito [19], there are always exactly 1- and 2-periodic wave solutions and if N ≥ 3, we need solve an over-determined nonlinear algebraic system to obtain a N-periodic wave. However, the situation of the coupled KdV–Toda-type equations is different. In this case, there are only exactly 1-periodic wave solutions and if N ≥ 2, it is necessary for us to deal with an over-determined algebraic system to solve N-periodic wave solutions (see §2 for details).

The paper is organized as follows. In §2, we will review the conditions given by Nakamura and apply them to the coupled KdV–Toda-type bilinear equations (1.1) and (1.2). In §3, we will propose a numerical procedure by using the Gauss–Newton method based on the condition. Section 4 devotes to some numerical results of the seven coupled KdV–Toda-type equations. Some conclusions and discussions will be given in §5.

2. Condition for N-periodic wave solutions

Firstly, we review Riemann’s θ-function defined by

$$\begin{array}{r}\theta (\mathit{\eta };\mathit{s}|\mathit{\tau })=\sum _{m_1}\sum _{m_2}\cdots \sum _{m_N=-\mathrm{\infty }}^{\mathrm{\infty }}\exp [\text{i}\sum _{j=1}^N(m_j+s_j){\eta }_j-\frac{1}{2}\sum _{j,k=1}^N(m_j+s_j){\tau }_{j,k}(m_k+s_k)],\end{array}$$ 2.1

where η_j, s_j and τ_j,k are the elements of the vector η, s and the symmetric matrix τ, respectively, and η_j is defined by

$${\eta }_j={\omega }_{j}t+k_{j}x+\cdots +{\eta }_j^0,j=1,2,\dots N.$$ 2.2

Here k_j, ω_j, ${\eta }_j^0$, the diagonal elements τ_jj and off-diagonal elements τ_jk(j ≠ k) are parameters corresponding to the wavenumbers, the frequencies, the phase positions, the amplitudes and the interactions, respectively.

(a). Single KdV-type bilinear equations

For a single KdV-type bilinear equation

$$F(D_t,D_x,\dots ,\lambda )\hspace{0.17em}f\cdot f=0,$$ 2.3

the condition for having N-periodic wave solutions was first proposed by Nakamura [1].

Lemma 2.1. —

For Riemann’s θ-function defined by (2.1), θ(η;0|τ) is a N-periodic wave solution of the single bilinear equation (2.3) if

$$\begin{array}{rl}\hspace{0.17em}& \sum _{m_1}\sum _{m_2}\cdots \sum _{m_N=-\mathrm{\infty }}^{\mathrm{\infty }}F[2i\sum _{j=1}^N(m_j-{\mu }_j/2){\omega }_j,2i\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})k_j,\dots ,\lambda ]\\ \hspace{0.17em}& \times \exp [-\sum _{j,k=1}^N(m_j-\frac{{\mu }_j}{2}){\tau }_{j,k}(m_k-\frac{{\mu }_k}{2})]=0,\end{array}$$ 2.4

for all possible combinations μ₁ = 0, 1, μ₂ = 0, 1, …, μ_N = 0, 1.

The proof of this lemma can be found in [1,19,34].

Note that there are 2^N equations of type (2.4), and the total number of parameters ω_j, k_j(j = 1, 2, …, N), λ, and τ_j,k(1 ≤ j, k ≤ N) is 2N + 1 + N(N + 1)/2. Generally, the diagonal elements τ_jj which influence the amplitudes, and the wave numbers k_j(or frequencies ω_j) are taken to be given parameters. Thus we have 2^N equations with 1 + N(N + 1)/2 unknowns. In the case of N = 1, 2, we have the equal number of equations and unknown parameters while in the case of N ≥ 3, the number of equations is larger than the number of unknown parameters, which means that this is an over-determined system.

(b). Coupled KdV–Toda-type bilinear equations

We apply lemma 2.1 to coupled KdV–Toda-type bilinear equations (1.1) and (1.2). Note that, there are an auxiliary variable z and two discrete variable m, n in this system. Therefore, the η_j in Riemann’s θ-function (2.1) is defined by

$${\eta }_j={\omega }_{j}t+l_{j}z+k_{j}x+p_{j}m+q_{j}n\cdots +{\eta }_j^0,j=1,2,\dots N.$$ 2.5

We have the following theorem.

Theorem 2.2. —

For Riemann’s θ-function defined by (2.1), (2.5), θ(η;0|τ) is a N-periodic wave solution of the coupled bilinear equations (1.1) and (1.2) if

$$\begin{array}{rl}\hspace{0.17em}& \sum _{m_1}\sum _{m_2}\cdots \sum _{m_N=-\mathrm{\infty }}^{\mathrm{\infty }}{F}_1[2i\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2}){\omega }_j,2i\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})l_j,\\ \hspace{0.17em}& 2i\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})k_j,i\sin \left(2{\delta }_1\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})p_j\right),i\sin \left(2{\delta }_2\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})q_j\right),\cdots ,c_1]\\ \hspace{0.17em}& \times \exp [-\sum _{j,k=1}^N(m_j-\frac{{\mu }_j}{2}){\tau }_{j,k}(m_k-\frac{{\mu }_k}{2})]=0,\end{array}$$ 2.6

and

$$\begin{array}{rl}\hspace{0.17em}& \sum _{m_1}\sum _{m_2}\cdots \sum _{m_N=-\mathrm{\infty }}^{\mathrm{\infty }}{F}_2[2i\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2}){\omega }_j,2i\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})l_j,\\ \hspace{0.17em}& 2i\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})k_j,i\sin \left(2{\delta }_1\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})p_j\right),i\sin \left(2{\delta }_2\sum _{j=1}^N(m_j-\frac{{\mu }_j}{2})q_j\right),\cdots ,c_2]\\ \hspace{0.17em}& \times \exp [-\sum _{j,k=1}^N(m_j-\frac{{\mu }_j}{2}){\tau }_{j,k}(m_k-\frac{{\mu }_k}{2})]=0,\end{array}$$ 2.7

for all possible combinations μ₁ = 0, 1, μ₂ = 0, 1, ·· ·, μ_N = 0, 1.

The proof of this theorem is similar to that of lemma 2.1 which is to substitute θ(η;0|τ) into the coupled bilinear system (1.1) and (1.2) and simplify the formula with some bilinear identities and tedious calculations. We omit the details here.

Note that there are 2^N+1 equations, and the total number of parameters ω_j, l_j, k_j(j = 1, 2, …, N), c₁, c₂, and τ_j,k(1 ≤ j, k ≤ N) is 3N + 2 + N(N + 1)/2. With k_j and τ_jj given, we obtain a nonlinear algebraic system of 2^N+1 equations with N + 2 + N(N + 1)/2 unknowns. Thus, for 1-periodic waves, we need to solve four parameters from four equations while for 2- and 3-periodic waves, we have to solve 7 and 11 parameters from 8 and 16 equations, respectively.

3. Numerical scheme

In this section, we will introduce our numerical procedure to solve the unknown parameters from the nonlinear algebraic system (2.6) and (2.7). The main idea is to formulate the problem as a nonlinear least-square problem and then use the Gauss–Newton method [35] to solve it.

For simplicity, we rewrite equations (2.6) and (2.7) as

$$\mathit{H}(\mathit{x})={(H_1,H_2,\dots ,H_{2^{N+1}})}^{\text{T}}=0,$$ 3.1

where H_i = 0(i = 1, 2, …, 2^N+1) is one of the equations in system (2.6) and (2.7) and x is a vector whose elements are the unknown parameters ω_j, l_j, τ_ij(i < j) and c₁, c₂. The objective function of the nonlinear least-square problem is

$$S(\mathit{x})=\frac{1}{2}\sum _{n=1}^{2^{N+1}}{H}_n^2(\mathit{x})=\frac{1}{2}\mathit{H}{(\mathit{x})}^{\text{T}}\mathit{H}(\mathit{x}),$$ 3.2

Starting with an initial guess x^[0], the Gauss–Newton method proceeds by the iterations

$${\mathit{x}}^{[j+1]}={\mathit{x}}^{[j]}-{({\mathit{J}}^{\mathsf{T}}\mathit{J})}^{-1}{\mathit{J}}^{\mathsf{T}}\mathit{H}{\mid }_{\mathit{x}={\mathit{x}}^{[j]}},$$ 3.3

where x^[j] (j ≥ 1) is the jth iterative output and J is the Jacobian matrix of H, i.e.

$$\mathit{J}={\left[\frac{\mathrm{\partial }{H}_i}{\mathrm{\partial }{x}_j}\right]}_{i=1,\dots ,2^{N+1};j=1,\dots ,N+2+N(N+1)/2}.$$ 3.4

This iterate process makes the objective function S(x) decay to zero. In the numerical experiments, if ${\mathit{J}}^{\mathsf{T}}\mathit{J}$ is near singular, change it to ${\mathit{J}}^{\mathsf{T}}\mathit{J}+{10}^{-6}\mathit{E}$ to modify the singularity, where E is the unit matrix.

The key of the procedure is the choice of initial guess x^[0]. We suggest the following guidance to determine the initial guess. For given k_j, solve the initial guess ${\omega }_j^{[0]}$ and $l_j^{[0]}$ from the equations

$$F_1(i{\omega }_j^{[0]},i{l}_j^{[0]},i{k}_j,\dots ,c_1^{[0]})=0$$ 3.5

and

$$F_2(i{\omega }_j^{[0]},i{l}_j^{[0]},i{k}_j,\dots ,c_2^{[0]})=0,$$ 3.6

where $c_1^{[0]}$, $c_2^{[0]}$ are the initial guess of c₁, c₂, and are generally taken to be 0, ± 1, ± 2. In fact, if the initial guess x^[0] satisfies equations (3.5) and (3.6), the objective function S(x) will have a smaller initial value.

4. Numerical results

In this section, we use the above numerical scheme to calculate 3-periodic wave solutions of several coupled KdV–Toda-type equations, including the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the BM lattice, the semi-discrete KdV equation, the rToda lattice and the (2+1)-dimensional Leznov lattice. All computations are carried out on a computer with a 2.83 GHz CPU and 8 GB main memory. The termination condition for stopping the numerical iteration is ||x^[j+1] − x^[j]||₂ < 10⁻¹⁴ and ||H||₂ < 10⁻¹⁴, where $||\cdot |{|}_2$ means 2-norm.

(a). Coupled Ramani equation

The coupled Ramani equations (1.5) and (1.6) were first proposed in [32], and their N-solitons were known to be expressed by Pfaffians [36] when the integral constants c₁ and c₂ are taken to be zero. Some other properties and generalizations can be found in [37–40]. As far as we know, there are no results about the periodic waves of this equation. It is worth mentioning that when v = 0, the coupled Ramani equation reduces to the following Ramani equation [41]:

$$u_{xxxxxx}+15u_{xx}{u}_{xxx}+15u_x{u}_{xxxx}+45u_x^2{u}_{xx}-5(u_{xxxt}+3u_{xx}{u}_t+3u_x{u}_{xt})-5u_{tt}=0.$$ 4.1

Note that there are two constants u₀ and v₀ in the variable transformation (1.9) and u₀ makes no difference to the bilinear equations while v₀ does. Thus the numerical experiments will be carried out with v₀ = 0 and v₀ = 1, respectively. When plotting the profile of u and v, we will take u₀ = 0, ${\eta }_j^0=0$ and z = 0 without loss of generality.

Some detailed numerical examples are given in table 1 (v₀ = 0) and table 2 (v₀ = 1). As shown in figures 1–4, the 3-periodic wave represents three waves interacting with each other repeatedly. The difference between the first two examples in the two tables is the wavenumbers k_i. The wavenumbers are chosen to be in proportion in the first example, thus the spatial period of the wave train is L = 10. However, in this case, the frequencies may not be in proportion. Hence, the first wave is periodic in space but quasi-periodic in time. The second wave is quasi-periodic both in the spatial and temporal directions. In the third example, we choose some large diagonal elements τ_ii which lead to small amplitudes. The corresponding numerical results show that c₁ and c₂ tend to zero which means the third wave approaches 3-solitons.

Figure 2.

The third example of the coupled Ramani equation in table 1. (a) u-profile; (b) contour plot of u. (Online version in colour.)

Figure 3.

The first example of the coupled Ramani equation in table 2. (a) u-profile; (b) contour plot of u; (c) v-profile; (d) contour plot of v. (Online version in colour.)

Table 1.

3-periodic waves to the coupled Ramani equation: examples with v₀ = 0.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.67 × 2π	0.86 × 2π	1.02 × 2π	1	1	0.4685	−0.8643
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
7.0815	−0.9501	1.0718	0.0183	−1.4992	1.0605	1.6167	24.5355	0.1485
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.67 × 2π	0.86 × 2π	1.02 × 2π	1	1	1.1568	1.9536
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
3.5583	0.0146	−0.0146	−0.0000	−2.5593	2.0866	3.0502	58.7556	−0.0000
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	1	1	0.2904	−0.3390
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−1.1440	−0.0000	0.0000	−0.0000	2.7731	2.1719	5.1502	0.0000	0.0000

Table 2.

3-periodic waves to the coupled Ramani equation: examples with v₀ = 1.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.67 × 2π	0.86 × 2π	1.02 × 2π	1	1	1.4388	3.1394
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
7.9404	1.6866	2.0555	1.5475	2.1251	1.2939	2.7630	24.0624	0.9121
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.67 × 2π	0.86 × 2π	1.02 × 2π	−1	1	−0.6767	2.3019
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−2.9109	−2.2818	2.1357	5.6401	0.2632	−0.9164	1.9118	11.1843	1.7826
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	1	1	1.4071	3.3919
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−2.9249	1.4311	1.7623	5.6511	2.2460	1.1912	1.9028	0.0000	0.0000

Figure 1.

The first example of the coupled Ramani equation in table 1. (a) u-profile; (b) contour plot of u; (c) v-profile; (d) contour plot of v. (Online version in colour.)

Figure 4.

The third example of the coupled Ramani equation in table 2. (a) u-profile; (b) contour plot of u. (Online version in colour.)

Remark 4.1. —

In what follows, we always give three numerical examples for each equation without repeat illustration.

In some cases, the numerical experiments will produce results with l_j = 0 (see the third example in table 1). As we stated before, this kind of solution will reduce to the solutions of the Ramani equation (4.1).

(b). Coupled KdV equation

The Hirota–Satsuma coupled KdV equation [42]

$$\begin{array}{rl}\hspace{0.17em}& u_t=\frac{1}{4}{u}_{xxx}+3u{u}_x+3{(-{\varphi }^2+\omega )}_x,\end{array}$$ 4.2

$$\begin{array}{rl}\hspace{0.17em}& {\varphi }_t=-\frac{1}{2}{\varphi }_{xxx}-3u{\varphi }_x\end{array}$$ 4.3

$$\begin{array}{r}\text{and}{\omega }_t=-\frac{1}{2}{\omega }_{xxx}-3u{\omega }_x.\end{array}$$ 4.4

can be transformed into the bilinear form

$$(D_x{D}_t-\frac{1}{4}{D}_x^4-\frac{3}{4}{D}_z^2+c_1)f\cdot f=0$$ 4.5

and

$$(D_z{D}_t+\frac{1}{2}{D}_z{D}_x^3+4c_2)f\cdot f=0,$$ 4.6

by the dependent variable transformation

$$u={(\ln f)}_{xx},\varphi =\frac{1}{2}{(\ln f)}_z+c_{2}t,\omega =\frac{1}{2}{\varphi }_z+{\varphi }^2,$$ 4.7

where z is an auxiliary variable, c₁ and c₂ are arbitrary constants. Its N-solitons (with c₁ = c₂ = 0) and infinitely many conserved quantities were also given in [42]. If we take ω = 0, equations (4.2)–(4.4) can reduce to a system of equations describing the interaction of two long waves with different dispersion relations proposed in [43]. Here we consider the 3-periodic waves to equations (4.2)–(4.4), and some detailed numerical examples are shown in table 3 and figures 5 and 6.

Table 3.

3-periodic waves to the coupled KdV equation.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.53 × 2π	0.75 × 2π	1.13 × 2π	1	1	0.4049	0.0701
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
3.4666	0.1940	1.0286	−2.6391	−2.3303	0.6676	0.7780	3.2020	−0.9733
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.53 × 2π	0.75 × 2π	1.13 × 2π	−1	1	0.3965	0.0667
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−0.5881	0.1151	1.1263	0.9768	−2.2894	0.9814	2.1552	1.9683	−0.7032
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	−1	1	1.9379	−7.7516
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−26.1615	2.4674	0.0000	0.0000	1.6094	1.2238	3.2189	0.0000	−0.0000

Figure 5.

The first example of the coupled KdV equation in table 3. (a) u-profile; (b) contour plot of u. (Online version in colour.)

Figure 6.

The third example of the coupled KdV equation in table 3. (a) u-profile; (b) contour plot of u. (Online version in colour.)

(c). Coupled Ito equation

Motivated by Hirota and Satsuma’s results on the coupled KdV equation, a coupled Ito system was given in [44]

$$\begin{array}{rl}\hspace{0.17em}& u_t=v_x,\end{array}$$ 4.8

$$\begin{array}{rl}\hspace{0.17em}& v_t=-2(v_{xxx}+3u{v}_x+3v{u}_x)-12\omega {\omega }_x+6p_x,\end{array}$$ 4.9

$$\begin{array}{rl}\hspace{0.17em}& {\omega }_t={\omega }_{xxx}+3u{\omega }_x\end{array}$$ 4.10

$$\begin{array}{r}\text{and}{p}_t=p_{xxx}+3u{p}_x,\end{array}$$ 4.11

which can be reduced to the Ito equation [45] when p = 0 and ω = 0. By the dependent variable transformation

$$u=2{(\ln f)}_{xx},v=2{(\ln f)}_{xt},\omega ={(\ln f)}_z+c_{1}t,p={\omega }^2+{\omega }_z,$$ 4.12

the coupled Ito equation (4.8)–(4.11) can be transformed into

$$(D_z{D}_t-D_x^3{D}_z+2c_1)f\cdot f=0$$ 4.13

and

$$(D_t^2+2D_x^3{D}_t-3D_z^2-c_2)f\cdot f=0,$$ 4.14

where z is an auxiliary variable and c₁, c₂ are arbitrary constants. The bilinear system (4.13) and (4.14) admits N-soliton solutions when c₁ = c₂ = 0. Here we give some detailed numerical examples on the 3-periodic wave solutions in table 4 and figures 7 and 8.

Table 4.

3-periodic waves to the coupled Ito equation.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.53 × 2π	0.75 × 2π	1.13 × 2π	−2	−2	0.4296	−0.5977
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−0.4586	−0.4296	0.5977	0.4586	−0.6286	−1.7156	3.6439	0.0566	0.2266
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.53 × 2π	0.75 × 2π	1.13 × 2π	2	2	−0.0396	0.3857
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
0.0434	0	0	0	1.1511	−1.8009	1.4570	0	−0.1087
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	2	2	0.4961	3.9688
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
0.0000	0	0	0	1.3499	1.9459	2.5649	0.0000	0.0000

Figure 7.

The first example of the coupled Ito equation in table 4. (a) u-profile; (b) contour plot of u; (c) v-profile; (d) contour plot of v. (Online version in colour.)

Figure 8.

The third example of the coupled Ito equation in table 4. (a) u-profile; (b) contour plot of u; (c) v-profile; (d) contour plot of v. (Online version in colour.)

(d). Blaszak–Marciniak lattice

The BM lattice (1.10)–(1.12) was first derived as an application of r-matrix formalism to the algebra of shifts operators. Some research on the BM lattice has been conducted [33,46], and its Bäcklund transformation and N-solitons (with c₁ = 2, c₂ = 0) were given in [47]. In this subsection, we are going to calculate the 3-periodic wave solutions to the BM lattice.

According to theorem 2.2, Riemann’s θ-function θ(η;0|τ) gives the 3-periodic wave solution to the coupled bilinear system (1.13) and (1.14). Here η is defined by

$${\eta }_j={\omega }_{j}t+k_{j}n+l_{j}z+{\eta }_j^0,j=1,2,3.$$

We give three examples in table 5, and plot the first and third examples with z = 0 and ${\eta }_j^0=0$ in figure 9.

Table 5.

3-periodic waves to the BM lattice.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.46 × 2π	1.02 × 2π	1.53 × 2π	−2	3	1.1286	2.2572
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
1.4438	0.5953	1.1907	2.1584	4.0631	0.7348	1.4696	2.5746	1.7671
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.46 × 2π	1.02 × 2π	1.53 × 2π	1	−2	0.7985	0.6917
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
1.1416	0.5027	0.4348	0.7862	2.4872	2.2088	1.9033	2.0224	0.0688
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	−2	3	0.7658	1.3801
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
1.7077	0.4988	1.0014	1.5331	2.1051	1.2391	2.9130	2.0000	0.0000

Figure 9.

Figures of 3-periodic waves to the BM lattice with different choice of n = 0, 3, 6, 9. (a) first example of table 5; (b) third example of table 5. (Online version in colour.)

(e). Semi-discrete KdV equation

The semi-discrete KdV equation given by Ohta & Hirota [48]

$$\frac{4}{1+a^2{u}_n}\frac{d}{dt}{u}_n={\mathrm{\Delta }}^{3}M{u}_n+6u_n\mathrm{\Delta }M{u}_n+a^2[\mathrm{\Delta }M(u_n{\mathrm{\Delta }}^2{u}_n)+3(\mathrm{\Delta }M{u}_n)({\mathrm{\Delta }}^2{u}_n)],$$ 4.15

where a is the discrete step, Δ and M are the difference and averaging operators defined by Δf_n = (f_n+a/2 − f_n−a/2)/a and Mf_n = (f_n+a/2 + f_n−a/2)/2, can be reduced to the KdV equation in the continuum limit a → 0. Through the variable transformation

$$u_n=\frac{1}{a^2}(\frac{f_{n+2}{f}_{n-1}}{f_{n+1}{f}_n}-1),$$ 4.16

the semi-discrete KdV equation (4.15) can be transformed into the bilinear form

$$[2a{D}_z\sinh \left(\frac{D_n}{2}\right)-\cosh \left(\frac{3D_n}{2}\right)+(c_1+1)\cosh \left(\frac{D_n}{2}\right)]f_n\cdot f_n=0,$$ 4.17

and

$$\begin{array}{r}[8a^3{D}_t\sinh \left(\frac{D_n}{2}\right)-2a{D}_z\sinh \left(\frac{3D_n}{2}\right)+(c_2-3)\cosh \left(\frac{D_n}{2}\right)+3(1-c_1)\cosh \left(\frac{3D_n}{2}\right)]f_n\cdot f_n=0,\end{array}$$ 4.18

where z is an auxiliary variable and c₁, c₂ are arbitrary constants. Bilinear equations (4.17)–(4.18) admit N-soliton solution when c₁ = c₂ = 0. As for the 3-periodic wave solutions, we give some detailed numerical examples in table 6, and plot the 3-periodic waves corresponding to the first and third examples in figure 10 with z and ${\eta }_j^0$ taken to be zero.

Table 6.

3-periodic waves to the semi-discrete KdV equation.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.46 × 2π	1.02 × 2π	1.53 × 2π	−2	3	0.0395	−0.2737
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−0.5759	0.6413	0.9640	0.9534	1.6223	0.8906	2.3114	0.0464	0.2523
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.46 × 2π	1.02 × 2π	1.53 × 2π	1	−2	0.0377	−0.1808
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−0.5959	0.6413	0.9012	0.9271	1.9047	0.8233	1.6944	0.0435	0.2445
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	−2	3	−0.0561	−0.3286
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
−0.6225	0.5878	0.9511	0.9511	1.9248	0.9624	2.3487	0.0000	0.0000

Figure 10.

(a) First example of the semi-discrete KdV equation in table 6; (b) third example. (Online version in colour.)

(f). Relativistic Toda lattice

In this subsection, we give some numerical examples of the relativistic Toda lattice [49,50]

$$\frac{\text{d}}{\text{d}t}{u}_n=u_n(v_{n-1}-v_n)$$ 4.19

and

$$\frac{\text{d}}{\text{d}t}{v}_n=v_n(v_{n-1}-v_{n+1}+u_n-u_{n+1}).$$ 4.20

Through the dependent variable transformation

$$u_n=\frac{\text{d}}{\text{d}t}\ln \frac{G_{n-1}}{F_n}+1\text{and}{v}_n=\frac{\text{d}}{\text{d}t}\ln \frac{F_n}{G_n}+1,$$ 4.21

and introducing an additional discrete variable m subject to

$$F_n=f_{m+\frac{1}{2},n}\text{and}{G}_n=f_{m-\frac{1}{2},n},$$ 4.22

then we obtain the following coupled bilinear form

$$[D_t\sinh (\frac{1}{2}{D}_n+\frac{1}{2}{D}_m)-\cosh (\frac{1}{2}{D}_n+\frac{1}{2}{D}_m)+c_1\cosh (-\frac{1}{2}{D}_n+\frac{1}{2}{D}_m)]f_{m,n}\cdot f_{m,n}=0$$ 4.23

and

$$[D_t\sinh \left(\frac{1}{2}{D}_m\right)+\cosh \left(\frac{1}{2}{D}_m\right)-c_2\cosh (-D_n+\frac{1}{2}{D}_m)]f_{m,n}\cdot f_{m,n}=0,$$ 4.24

where c₁, c₂ are arbitrary constants. The bilinear Bäcklund transformation and soliton solutions (with c₁ = c₂ = 1) for the rToda lattice were given in [51].

According to theorem 2.2, the 3-periodic wave solution to the coupled bilinear system (4.23) and (4.24) has the form f_m,n = θ(η;0|τ) with

$${\eta }_j={\omega }_{j}t+k_{j}n+l_{j}m+{\eta }_j^0,j=1,2,3.$$

Some detailed numerical results are given in table 7, and the 3-periodic waves corresponding to the first and third example are shown in figure 11.

Table 7.

3-periodic waves to the rToda lattice.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.23 × 2π	0.46 × 2π	1.02 × 2π	2	0	0.2170	0.5819
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
1.1102	0.4697	0.8257	1.0175	−0.7230	−0.9247	2.2316	0.7910	0.9365
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.23 × 2π	0.46 × 2π	1.02 × 2π	0	0	0.2207	0.4345
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
0.9602	0.4725	0.7778	1.2618	1.1665	0.7497	1.6420	0.9346	0.9879
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	2	0	0.2651	0.5611
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
0.9258	0.4406	0.8573	1.2180	2.0346	1.1468	2.7430	1.0000	1.0000

Figure 11.

Figures of 3-periodic waves to the rToda lattice with $m=0,{\eta }_j^0=0$. (a) First example of table 7; (b) third example of table 7. (Online version in colour.)

(g). Leznov lattice

The Leznov lattice is a (2+1)-dimensional lattice system proposed in [52]

$$\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }x\mathrm{\partial }y}\ln u(n)=u(n+1)v(n+1)-2u(n)v(n)+u(n-1)v(n-1),$$ 4.25

and

$$\frac{\mathrm{\partial }v(n)}{\mathrm{\partial }y}=u(n+1)-u(n-1),$$ 4.26

which is a special case of the so-called UToda(m₁;m₂) system with m₁ = 1, m₂ = 2. By the dependent variable transformation

$$u(n)=\frac{f(n+1)f(n-1)}{f^2(n)}\text{and}v(n)=\frac{1}{2}\cdot \frac{D_x{D}_{y}f(n)\cdot f(n)}{f(n+1)f(n-1)},$$ 4.27

system (4.25) and (4.26) can be transformed into the following coupled bilinear form

$$[D_y{D}_z-2\cosh (D_n)+c_1]f(n)\cdot f(n)=0$$ 4.28

and

$$[D_x{D}_y-2D_z\sinh (D_n)+c_2\cosh (D_n)]f(n)\cdot f(n)=0,$$ 4.29

where z is an auxiliary variable and c₁, c₂ are arbitrary constants. The coupled bilinear system (4.28) and (4.29) can be viewed as the bilinear form of a two-dimensional generalization of the BM lattice if we choose x = y = t. In the case of c₁ = 2 and c₂ = 0, its N-soliton solution was given in [40,53].

For the 3-periodic wave solutions, we take η_j in Riemann’s θ-function (2.1) as

$${\eta }_j=k_{j}n+{\omega }_{j}x+v_{j}y+l_{j}z+{\eta }_j^0,j=1,2,3,$$

where k_j, v_j and τ_jj are given parameters. Some numerical examples are shown in table 8. We plot the 3-periodic waves corresponding to the first and third examples in figures 12 and 13 with z and ${\eta }_j^0$ are chosen to be zero.

Table 8.

3-periodic waves to the (2+1)-dimensional leznov lattice.

k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	0.67 × 2π	0.86 × 2π	1.02 × 2π	1	2	0.2905	0.4218
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
0.3457	0.3085	0.5579	0.7016	1.9404	1.2144	2.7887	2.0408	0.0521
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$1.73\times \frac{2\pi }{10}$	$3.12\times \frac{2\pi }{10}$	0.67 × 2π	0.86 × 2π	1.02 × 2π	2	2	0.2979	0.3042
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
0.3533	0.3108	0.4320	0.7392	1.9566	1.1556	2.2070	2.0382	0.0450
k1	k2	k3	τ11	τ22	τ33	$c_1^{[0]}$	$c_2^{[0]}$	ω1	ω2
$1\times \frac{2\pi }{10}$	$2\times \frac{2\pi }{10}$	$3\times \frac{2\pi }{10}$	5.46 × 2π	5.02 × 2π	5.53 × 2π	2	2	0.2844	0.4162
ω3	l1	l2	l3	τ12	τ13	τ23	c1	c2
0.3504	0.3040	0.5499	0.6945	2.1085	1.2518	2.9479	2.0000	0.0000

Figure 12.

The first example of the (2+1)-dimensional Leznov lattice with n = 1 in table 8. (a) u- and v-profile; (b) contour plot of u and v. (Online version in colour.)

Figure 13.

The third example of the (2 + 1)-dimensional Leznov lattice with n = 1 in table 8. (a) u- and v-profile; (b) contour plot of u and v. (Online version in colour.)

5. Conclusion and discussion

A numerical process of calculating N-periodic waves to the coupled KdV–Toda-type equation is presented and some numerical experiments are carried out with the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and the relativistic Toda lattice. The numerical results show that the process is efficient in calculating 3-periodic waves. Here we give two remarks. Firstly, the numerical results are not unique since the system we solved is nonlinear and over-determined. Secondly, the third example for each equation shows that if we choose some large diagonal elements of the Riemann matrix which means the amplitudes tend to zero, then the corresponding 3-periodic wave solution go to the 3-soliton solution.

Data accessibility

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Competing interests

We declare we have no competing interests.

Funding

This work was partially supported by the National Natural Science Foundation of China(grant nos. 11571358, 11931017, 11871336, 12071447, 11971473, 11871444 and 11731014) and Fundamental Research Funds for the Central Universities(grant no. 201964008).