Inverse Spectral Transform for the Modified Kadomtsev‐Petviashvili Equation

Abstract

The 2 + I-dimensional modified Kadomtsev-Petviashvili (mKP) equation is studied by the inverse-spectral-transform method. The initial-value problems for the mKP-I and mKP-II equations are solved by the nonlocal Riemann-Hilbert and a-problem techniques for initial data decaying sufficiently rapidly at infinity. The lump solutions for the mKP-I equation are found explicitly. Wide classes of the exact solutions for the mKP equation-namely, the rational solutions, including the plane lumps for the mKP-I equation; solutions with functional parameters; the plane solitons; and breathers-are constructed by the use of the a-dressing method based on the nonlocal a-problem. The Miura transformation between the mKP and KP equations is discussed.

Conservation Laws, Symmetries, and Line Soliton Solutions of Generalized KP and Boussinesq Equations with p-Power Nonlinearities in Two Dimensions

“…with D being a Hamiltonian operator [20], and with the Hamiltonian density now being a nonlocal expression in terms of u, w. Note that if we put σ 2 = 0, then we obtain a Hamiltonian formulation of the 1D gB equation, which coincides in the case p = 1 with the first Hamiltonian structure [20] of the ordinary Boussinesq equation (5). These Hamiltonian formulations motivate studying the symmetries and conservation laws of the gKP and 2D gB equations in their respective potential forms (9) and (14).…”

“…We will first consider point symmetries (x, y, t, v) → (x,ỹ,t,ṽ), wherex,ỹ,t,ṽ are functions of x, y, t, v. For the gKP and 2D gB equations (9) and (14), an infinitesimal point symmetry consists of a generator…”

“…holding for all solutions v(x, y, t) of the gKP potential equation. Similarly, invariance of the 2D gB equation in potential form (14) is given by…”

“…(ii) No additional conservation laws corresponding to variational point symmetries admitted by the 2D generalized Boussinesq potential equation (14) arise for any p = 0.…”

Discrete symmetry’s chains and links between integrable equations

“…In Sec. 6 we develop a technique for constructing of exact localized solution of DS equations with a reduction constraint imposed and give some of these solutions. We conclude with a discussion on possible role of discrete symmetry's dressing chains in the theory of integrable PDE.…”