Interactions among different types of nonlinear waves described by the Kadomtsev–Petviashvili equation

Abstract

In nonlinear physics, the interactions among solitons are well studied thanks to the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlevé waves. In this paper, the nonlocal symmetries related to the Darboux transformations (DT) of the Kadomtsev-Petviashvili (KP) equation is localized after imbedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions such that interaction solutions among different types of nonlinear waves can be found. It is shown that starting from a Boussinesq wave or a KdV-type wave, which are two basic reductions of the KP equation, the essential and unique role of the DT is to add an additional soliton.

Interactions between solitons and other nonlinear Schrödinger waves

“…It should be mentioned that the tsunami waves may be described by other soliton equations such as the Korteweg de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations. The similar soliton-cnoidal wave interaction solutions and the same conclusions can be obtained [16,15]. In optics, especially for the fiber solitons, the NLS equation possesses the space variable t and the time variable x.…”

“…Recently, it is found that combining the symmetry reduction method and the DT or BT related nonlocal symmeries [14], one can readily find the interaction solutions among solitons and other nonlinear excitations including the cnoidal waves for the KdV [15] and KP [16] equations. In this paper, much simpler while more effective methods are developed for the NLS system while they are also valid for any other integrable systems.…”

“…The group invariant solutions (34) and (36) hint us that we may develop a simpler while effective method to solve the AKNS system as for other nonlinear systems such as the KdV and KP equations [16]. It is noted that the exact solutions of the reduction systems (35) and (37) are left to be discussed at the end of the next section.…”

Consistent Riccati Expansion for Integrable Systems

“…Some special interaction solutions for the KdV and KP systems are plotted in Figures – . Some other interaction solutions between solitons and cnoidal waves for the KdV equation, the KP equation, and the AKNS/NLS system have been discussed in several references . From Figures – 4 of this paper, we can see that the interaction between the soliton and every peak of the periodic wave is elastic with explicit phase shifts.…”

“…It will be seen later that the interaction between a soliton and a cnoidal wave for the w wave is another new universality for many CRE solvable systems, the expression with and exists for various integrable systems including the celebrate KdV, KP, NLS/AKNS, sG, SK, KK, Boussinesq, (modified) ANNV, etc. Some detailed special solutions of the form have been obtained by the nonlocal symmetry related symmetry reductions and one special soliton–cnoidal wave interaction solution is graphically displayed in . For completeness and to see the interaction property more clearly, we write down a much simpler interaction solution for the w equation by fixing the parameters , and ω 1 as …”