Interaction Behaviours Between Soliton and Cnoidal Periodic Waves for the Cubic Generalised Kadomtsev–Petviashvili Equation

1 Introduction

The reductive perturbation method is a powerful and efficient way of deriving the models describing nonlinear wave propagation and interaction [1]. By using the reductive perturbation method, a cubic generalised Kadomtsev–Petviashvili (CGKP) equation [2] for ultrashort spatiotemporal optical pulse propagation in cubic (Kerr-like) media is derived from the Maxwell–Bloch equations [3]. Contrary to the KP equation, the CGKP equation is not an integrable system. It adds lots of difficulties for finding the exact soliton solutions, in particularly the interaction solutions among solitons and other kinds of complicated waves. Thus, how to find these exact solutions is an important problem [4–7]. Recently, a consistent tanh expansion (CTE) method has been proposed to identify CTE solvable systems [8, 9]. The interaction solutions between one soliton and the cnoidal waves can be found with the CTE method. The method is valid for various integrable nonlinear systems, including the nonlinear Schrödinger system [10], the Broer–Kaup system [11], the higher-order KdV equation [12], and so on [13–15]. So far, the CTE method has been successfully applied to the integrable nonlinear systems. The method for the non-integrable, nonlinear systems has not yet been studied. In this article, we use the CTE method to study the non-integrable CGKP system.

This article is organised as follows. In Section 2 we prove that the CGKP equation fails to pass the Painlevé test with the standard Weiss–Tabor–Carnevale (WTC) method. Then, we devote the CTE approach to the CGKP equation. Although the CGKP is not a CTE solvable system, it is still useful in obtaining the exact interaction solutions. The explicit analytic interaction solutions between one kink soliton and the cnoidal waves are investigated through the CTE method. Section 3 is a simple summary and discussion.

2 Painlevé Analysis and New Interaction Solutions for the CGKP System

The CGKP equation reads [2]

(1)(ut + Au2ux + Buxxx)x − uyy = 0, (1)

where A and B are arbitrary constants. It describes the (2+1)-dimensional few-optical-cycle spatiotemporal soliton propagation in cubic nonlinear media [16, 17]. The collapse threshold for the propagation of ultrashort spatiotemporal pulses for the CGKP equation was calculated by the direct numerical method [3].

First, we applied the standard WTC approach [18] to test the Painlevé property of (1). According to the standard WTC method, we selected u as the following ansatz

(2)u = ∑j = 0∞ujϕj − α, (2)

where u_j is an arbitrary function of (x, t). From the leading order analysis, the following results were obtained:

(3)α = 1, u0 = −6BAfx. (3)

Substituting expressions (3) and (2) into (1) and comparing the coefficient of ϕ^j–5, the polynomial equation in j was derived as

(4)j4 − 10j3 + 29j2 − 8j − 48 = 0, (4)

Thus, the resonance values of j were found to be

(5)j = −1, 3, 4, 4. (5)

The resonance at j= –1 corresponded to an arbitrary singularity manifold ϕ(x, y, t)=0. In order to check the existence of a sufficient number of arbitrary functions at other resonance values, we proceeded further to the coefficients of ϕ^–4 and ϕ^–3, where the explicit values of u₁ and u₂ were obtained:

(6)u1 = −−3B2Aϕxxϕx, u2 = −16AB1ϕx 3(ϕxϕt − ϕy 2 + Bϕxϕxxx − 32Bϕxx 2). (6)

Similarly, gathering the coefficients of ϕ^–2 and ϕ^–1 and combining (6), we could easily verify that the resonance conditions were not satisfied. From the aforementioned considerations, we deduced that the CGKP equation is not a Painlevé integrable system. Although it failed to pass the Painlevé test, many interesting mathematical properties for the CGKP equation still need further study.

In this part, we studied some new types of exact interaction solutions of the CGKP system via the CTE method. For the CGKP system, we took the following truncated tanh function expansion form by using leading order analysis:

(7)u = u0 + u1tanh(f), (7)

where u₀, u₁, f are arbitrary functions of (t, x, y). By substituting (7) into the CGKP system (1) and vanishing coefficients of the powers of tanh(f)⁵ and tanh(f)⁴, we got

(8)u0 = −−3B2Afxxfx, u1 = −6BAfx. (8)

Collecting the coefficients of tanh(f)³, tanh(f)², tanh(f)¹ and tanh(f)⁰, we obtained the following four overdetermined systems:

(9)ftfx + Bfxfxxx − 32Bfxx2 − 2Bfx4 − fy2 = 0, (9)

(10)fxfyy + 2fyfxy − 2fxfxt − ftfxx    + B(5fxxfxxx − 2fxfxxxx + 14fx3fxx − 32fxx3fx) = 0, (10)

(11)fxxt + 2fxfy2 − 2ftfx2 − fxxy + B(fxxxxx + 4fx5 − 9fxfxx2 − 8fxxxfx2− 3(fxxfxxx)xfx + 152fxxxfxx2fx2 − 3fxx4fx3) = 0, (11)

(12)ftfxx + 2fxfxt − fxfyy − 2fyfxy + fxxyy − fxxt2fx + fxxfxxt − fxyfxxyfx2  + fxxxfxt − fxxfxxy2fx2 + fxxfxy2 − fxx2fxtfx3 + B(4fxxfxxx − 14fx3fxx + 5fxfxxxx − 12fxxxxxxfx + 32fxx3fx + 5fxxxfxxxxfx2 + 52fxxfxxxxxfx2 − 274fxxfxxx2fx3 − 374fxx2fxxxxfx3 + 994fxx3fxxxfx4 − 9fxx5fx5) = 0. (12)

Because of the fact that the CGKP equation is not an integrable system, the aforementioned four equations are not consistent. It brings us some troubles to solve the CGKP system (1) using the CTE method. Alhough the CGKP equation did not pass the CTE test, we also got the following nonauto-Bäcklund transformation (BT) theorem.

Nonauto-BT theorem. If one finds that the solution f can satisfy (9)–(12) simultaneously, then u with

(13)u = −6BAfxtanh(f) − −3B2Afxxfx, (13)

is a solution of the CGKP system (1).

According to nonauto-BT theorem, we obtained the exact solutions of the CGKP (1) by solving the overdetermined systems (9)–(12). Here are some interesting examples.

A quite trivial solution of (9)–(12) has the form

(14)f = k0x + l0y + ω0t, ω0 = 2Bk03 + l02k0, (14)

where k₀ and l₀ are the free constants and ω₀ is determined by the dispersion relations. Substituting the trivial solution (14) into (13), one kink soliton solution of CGKP system yields

(15)u = −6BAk0tanh(k0x + l0y + 2Bk04 + l02k0t). (15)

Thus, we can find some nontrivial solutions of the CGKP equation from some quite trivial solution of (14).

To find interaction solutions between one kink soliton and other nonlinear excitations, we assume the interaction solution form as

(16)f = k0x + l0y + ω0t + F(X),  X = kx + ly + ωt, (16)

where k₀, l₀, ω₀, k, l, and ω are all free constants. Substituting the expression (16) into (9)–(12), these four equations are consistent with satisfying the following relation:

(17)l0 = k0kl, (17)

and the consistent condition is

(18)FX 4 + 4k0kFX 3 + 12Bk2k02 − kω + l22Bk4FX 2  + 8Bk2k03 − k2ω0 − kk0ω + 2k0l22Bk5FX−12FXFXXX  + 34FXX2 − k02kFXXX + k1(2Bk2k03 − k2ω0 + k0l2)2Bk6 = 0, (18)

then we obtain an equation about F₁(X) as

(19)F1X  2 − 4F14 − a1F13 − a2F12 − a3F1 − a4 = 0, FX = F1, (19)

where

a1 = C1k3 + 12k0k, a2 = 3C1k2k0 + 12k02k2 + 2kω − 2l2Bk4,a3 = 3C1kk02 + 4k03k3 + k2ω0 + 3kk0ω − 4k0l2Bk5,a4 = C1k03 + k0(k2ω0 + kk0ω − 2k0l2)Bk6,

and C₁ is arbitrary constant. The general solution of (19) can be written out in terms of Jacobi elliptic functions [19]. Hence, the solution expressed by (13) is just the explicit exact interaction between one kink soliton and the cnoidal periodic waves. To show more clearly this kind of solution, we offer two special cases for solving (19).

Case 1. We take one solution of (19) as

(20)F1 = c0 + c1sn(c2X, m), (20)

which leads to the interaction solution between one kink soliton and the cnoidal wave solution of the CGKP system (1):

(21)u = −6BA(c1sn(c2X, m) + c0k + k0)tanh(c0(k + k0)x + l(c0k + k0)ky + (c0ω + ω0)t + k0c2m + c1mc2ln(dn(c2X, m) − mcn(c2X, m))) − −3B2Ak2c1c2cn(c2X, m)dn(c2X, m)c1ksn(c2X, m) + c0k + k0, (21)

where sn, cn, and dn are the usual Jacobian elliptic functions with the modulus m and

c1 = k0m4k − 116C1k3m, c2 = 18C1k3 − k02k,ω = kB(5 − m5)(C1k4 − 4k0)2128 + l2k,ω0 = B64k0k8C12(m2 + 1) − BC13512k12(m2 − 1) − BC132k02k4(m2 + 7) + B2k03 + k0l2k2,

Figure 1 plots the interaction solution between one kink soliton and the cnoidal wave with the parameters k=–1, k₀=1, l=1, c₂=–1, m=0.3, A=–2, B=6, C₁=–3. We can see that the field u exhibits one kink soliton propagated on the cnoidal wave background. Figure 2 plots the interaction solution with selecting m=0.8 and the other parameters same as Figure 1. It demonstrates that the interaction behaviour is different with selecting different parameters.

Figure 1:

Plot of one kink soliton on the cnoidal wave background expressed by (21) of the CGKP equation in one and two dimensions respectively: (a) one-dimensional image at t= y= 0; (b) one-dimensional image at t= x= 0; (c) two-dimensional image at t= 0.

Figure 2:

Plot of one kink-shaped soliton on the quasi-periodic cnoidal wave background expressed by (21) of the CGKP equation in one and two dimensions respectively: (a) one-dimensional image at t= y= 0; (b) one-dimensional image at t= x= 0; (c) two-dimensional image at t= 0.

Case 2. Another special solution of (19) reads

(22)F1 = b0b11 − b2sn(b1X, m)2, (22)

which leads to the interaction solution for the CGKP system (1):

(23)u = −6BAb2k0sn(b1X, m)2 − b0b1k − k0b2sn(b1X, m)2−1tanh(k0x + k0lky + ω0t + b0Eπ(sn(b1X, m), b2, m)) − −6BAk2b0b12b2 sn(b1X, m)cn(b1X, m)dn(b1X, m)(b2sn(b1X,m)2 − 1)(b2k0sn(b1X, m)2 − b0b1k − k0), (23)

where E_π is the third type of incomplete elliptic integral and

b2 = m2(b12k2m2 + k02 + k04 + b14k4m4 + 2b12k2k02(m2 − 2))2k02, b0 = b1kk0(m2b2 − 2m2 + b2)(b12k2m2 + k02b2),C1 = 4(b1k2m2 − b0b1b2kk0 − 2k02b2)k4k0b2, ω0 = 2Bk2(b0b13m2k3 + k03b2) + k0l2b2k2b2,ω = 2k0l2 − k2ω0 − BC1k6k02kk0,

Figure 3 plots interaction solutions between one kink soliton and the cnoidal wave expressed by (23) with the parameters k= 1, k₀=2, l=1, b₁=1, m=0.9, A=–2, B=6. In the ocean, there are some typical nonlinear waves, such as interaction solutions between the solitary waves and the cnoidal periodic waves [19]. Solutions (21) and (23) may be useful for studying the interactions between theses types of ocean waves.

Figure 3:

Plot of one kink soliton on the cnoidal wave background expressed by (21) of the CGKP equation in one and two dimensions respectively: (a) one-dimensional image at t= y= 0; (b) one-dimensional image at t= x= 0; (c) two-dimensional image at t= 0.

3 Conclusions

In summary, the CGKP equation was studied by means of the CTE method. Although the CGKP equation could not pass the CTE test, the CTE method is still useful for construction of the interaction solutions. We presented the interaction solutions between one kink soliton and the cnoidal waves using the CTE method. Some special types in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral were explicitly discussed both in analytical and graphical ways. Thus, the method is valid to the non-integrable model. Other methods, such as the tanh-function method [20, 21], the sub-equation method [22–24], the tri-function method [25], and the multiple exp-function method [26, 27] for the integrable and non-integrable systems have been used to obtain some traveling wave and multiple-wave solutions. The interaction solutions among solitons and other nonlinear excitations for the integrable nonlinear equations were also established with the truncated Painlevé expansion [8], the Darboux transformation [19, 28, 29], and the Bäcklund transformation [30] related nonlocal symmetries. In the meantime, some physical situation models in the study of nonlinear wave propagation were derived with the reductive perturbation method [1, 31]. The details on the solutions for these models are very promising for both the fundamental and applicative points of view.