Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

1 Introduction

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of nature science, especially nonlinear physics [1], [2], [3]. Since the source of a nonlinear problem is closely linked with reality, it contains rich in content and broad, and the corresponding nonlinear model is also complicated. Because of the complexity of the nonlinear system, there are still a lot of nonlinear equations for which the exact solutions are not easy to be obtained. Therefore, how to get exact solutions or more new accurate analytical solutions for a given nonlinear equation has become an important topic to scientists. In recent decades, many effective methods of solving differential equations have been constructed [2], [3], [4]. Among these methods, in 1989, Clarkson and Kruskal [5] proposed a simple direct method (CK direct method) which is different from the Lie group method, since it can find all the possible similarity reductions without using any group theory. Soon after, Lou optimised the above-mentioned approach and presented the improved CK method [6], [7]. Most recently, Lou and Ma [8] applied the direct method of symmetry transformation group for Lax integrable system in place of the traditional method of solving symmetry transformation group.

As we all know, the nonlocal symmetry has closed relation with the integrable model and is beneficial to the enlarge the class of the symmetry that provides the chance of obtaining the exact solution. However, the nonlocal symmetry canot be used to construct solution directly. In other words, only nonlocal symmetry is not enough unless we localised nonlocal symmetry into the local ones with the closed prolonged system. Very recently, from the nonlocal symmetry related to Darboux transformation, Lou et al. [9] obtained the explicit analytic interaction solutions between cnoidal waves and solitary wave for the well-known KdV equation. Furthermore, by studying the new exact solutions of the equations such as mKdV equation, ANKS system and Boussinesq equation, Xin [10], [11], [12] proved the effectiveness of this proposed method.

In this article, the structure of the layout is as follows: In Section 2, under the infinitesimal transformation of variable u, a vector symmetry which contains the classical Lie point symmetry and the nonlocal symmetry is derived directly. In Section 3, with the aid of the Lax pair of KdV equation, the nonlocal symmetry is localised in the properly prolonged system by introducing two suitable auxiliary variables. As a result, the finite transformation and the general Lie point symmetry of the prolonged system can be obtained. In Section 4, on the basis of the prolonged system, some new explicit solutions are constructed through similarity reductions. The different wave structure pictures give a visual representation of these solutions. Section 5 is a simple conclusion of this article.

2 Nonlocal Symmetry of the KdV Equation

The KdV equation in canonical form can be expressed as (1, 2)

which is widely regarded as a good model for the description of weakly nonlinear long waves in many branches of physics and engineering field. In (1), u≡u(x, t) is an appropriate field variable, t is time, and x is a space coordinate in the direction of propagation. This well-known model describes how waves evolve under the competing but comparable effects of weak nonlinearity and weak dispersion [3]. In fact, if it is supposed that x-derivatives scale as ϵ where ϵ is the small parameter characterising long waves (i.e. typically the ratio of a relevant background length scale to a wavelength scale), then the amplitude scales as ϵ² and the time evolution takes place on a scale of ϵ⁻³. Based on the Bell polynomials scheme [4], the Lax pair of KdV equation (1) has

under the compatibility condition ψ_xx,t=ψ_t,xx.

Under the transformation (with the infinitesimal parameter ε)

the symmetry σ₁ of the KdV equation (1) is a solution of the following linearised equation

Supposing the symmetry σ₁ with the auxiliary variables ψ and its one-order partial derivative ψ_x

then substituting (6) into (5) and solving the determining equations, the vector symmetry can be derived

where c_i(i=1, …, 5) are five arbitrary constants. Equation (7) contains the classical Lie point symmetry σ11 = (c1x2 − 6c3t − c5)ux + (3c1t2 − c4)ut + c1u + c3 and the nonlocal symmetry σ₁₂=c₂ψψ_xe^−2μt.

3 Localisation of the Nonlocal Symmetry

For the vector symmetry (7), letting c₁=c₃=c₄=c₅=0 and c₂=1, we have the nonlocal symmetry

At the same time, the linearised equations

are the direct results of (2) and (3), respectively, under the symmetry transformation ψ→ψ+εσ₂.

In order to obtain the localisation of the nonlocal symmetry (7), taking ψ₁=ψ_x and p = ∫ψ2dx, and the corresponding linearised equations are

under the transformation ψ₁→ψ₁+εσ₃ and p→p+εσ₄. The equation

is a direct result from the Lax pair of KdV equation (1).

Therefore, the linear system (5), (9)–(13) has a solution

The above-mentioned solution (14) indicates that the nonlocal symmetry is localised in the properly prolonged system with the Lie point symmetry vector of (14), namely

Using the following initial condition

the corresponding finite transformation reads

Here, {u, ψ, ψ₁, p} is a solution of the prolonged system (1) with ψ₁=ψ_x and p = ∫ψ2dx.

Under the transformation

the general Lie point symmetry of the prolonged system is

or has the following form

With the help of computer programs, such as Maple, the solution can be obtained

From (21), one can obtain the following six operators

Hence, we obtain the commutator table listed in Table 1 with the (i, j)-th entry indicating [M_i, M_j] according to the commutator operators [M_p, M_q]=M_pM_q−M_qM_p.

4 Similarity Reduction and the Wave Structures of the KdV Equation

Consider the following characteristic equation

where ξ, τ, U, Ψ, Ψ₁, and P are dedicated by (21). Without loss of generality, taking c1 = 0, c2 = −4k1, c3 = −A2 − c62k1, c4 = 1, c5 = k and μ=0, one has

where X=x−kt, A, k, k₁, and c₆ are arbitrary constants. Substituting (24) into the prolonged system, one can derive

and G(X) satisfies

One can simplify (26) using 1W(X) to replace G(X), and the reduced equation is

Equation (27) can be transformed to

or

where K3 = −A2k1, K2 = 12λ − 2k, K1 = c0, K0 = −k1A.

One know that the general solution of (29) can be written out in terms of Jacobi elliptic functions. Hence, the solution expressed by (24) is just the explicit exact interaction between the soliton and the cnoidal periodic wave.

We assume W(X)=a₀+a₂sn²(cX, m) as a simple solution of (29). Then

where c satisfies c2 = −a2K02a0(a02m2 + a0a2m2 + a0a2 + a22). Here the terms sn, cn, and dn are three usual Jacobian elliptic functions with modulus m, whereas a₀ and a₂ are two independent constants. The incomplete elliptic integral EllipticPi is defined by EllipticPi(z, ν, k) = ∫0z1(1 − νt2)1 − t21 − k2t2dt. For the term K₀, it is connected with K₁, K₂, K₃, and the relation reads as follows

Therefore, the formula of (24) can be rewritten

with

In order to study the structure of this solution, some figures which corresponds to soliton solutions found above are given. Figure 1 shows the expression (33–35), respectively. Figure 2 shows the evolution of intensity profile u of soliton solution (32) of KdV equation, where the parameters are fixed as {m, λ, a₀, a₂, K₀, K₁, K₂, K₃, c}={0.8, 1, 2, –0.3333, 1.7961, –22.9169, –13.4666, –2.6170, 0.5837}. One can see that the component u exhibits a soliton propagating on a cnoidal wave background. In fact, it is of interest to study such types of analytical solutions. As we know, solitary waves and cnoidal periodic waves are two typical nonlinear waves widely appearing in many physical fields such as ocean. Here we mainly devote to obtain the exact form of soliton–cnoidal waves solution from the original model equation. It is expected to the realistic physical interpretation and experiment observation. For instance, in a recent work [13], the oblique propagation of ion-acoustic soliton–cnoidal waves was reported in a magnetised electron-positron-ion plasma with superthermal electrons. For this kind of soliton–cnoidal waves solution, it is shown that every peak of a cnoidal wave elastically interacts with a usual soliton except for some phase shifts. The authors discussed the influence of the electron superthermality, positron concentration, and magnetic field obliqueness on the soliton–cnoidal wave in detail.

Figure 1:

Evolution of the wave distribution solution as a function of propagation distance x and time t. (a–c) show the corresponding variables u_1,u_2, and u₃ expressed by (33)–(35), respectively.

Figure 2:

Interaction graphs to the KdV equation. (a–c) show the time evolution view at t=–1, 0 and 1, respectively. (d) Evolution of the wave distribution solution (32) as a function of propagation distance x and time t.

5 Summary and Conclusion

Based on the original Lie symmetry method and some changes of the assumption form of the symmetry, we derive not only the local symmetry, but the nonlocal symmetry which is just one of the destinations of this article. With the introduction of two auxiliary variables ψ1 = ψx, p = ∫ψ2dx, we localise the nonlocal symmetry into the Lie point ones with the closed prolonged system successfully. Starting from the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are obtained. The physical interaction between the cnoidal waves and a solitary wave is illustrated through the image simulation.