In the process of searching for explicit solutions, quite a few systematic methods have been developed, such as inverse scattering transformation [1], Darboux transformations [2], Hirota’s bilinear method [3-5], and so on. Among them, the bilinear method first proposed by Hirota provides us with a comprehensive approach to construct exact solutions of nonlinear evolution equations (NEEs). Meanw...

Exact solutions of the (2+1) – dimensional Kadomtsev – Petviashvili by Zhang [Zhang H., Applied Mathematics and Computation 216 (2010) 2771 – 2777] are considered. To look for ”new types of exact solutions travelling wave solutions” of equation Zhang has used the G’/G – expansion method. We demonstrate that there is the general solution for the reduction by Zhang from the (2+1) – dimensional Ka...

Abstract The current work focuses on the solutions of Kadomtsev and Petviashvili (KP) equation, which models nonlinear waves in a dispersive medium. modified auxiliary equation approach is utilized to find analytical KP equation. Consequently, set including Jacobi elliptic solitary periodic obtained. geometry derived plotted with an appropriate choice parameters. It can be seen that proposed me...

The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrödinger/Gross–Pitaevskĭi (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion re...

Explicit solitary waves are known to exist for the Kadomtsev-Petviashvili-I (KP-I) equation in dimension 2. We first address numerically the question of their Morse index. The results confirm that the lump solitary wave has Morse index one and that the other explicit solutions correspond to excited states. We then turn to the 2D Gross-Pitaevskii (GP) equation which in some long wave regime conv...

The Krichever-Novikov equation u t = u xxx − 3 2u x (u 2 xx − r(u)) + cu x , r (5) = 0 (1) appeared (up to change u = p(˜ u), ˙ p 2 = r(p)) in [1] for the first time in connection with study of finite-gap solutions of the Kadomtsev-Petviashvili equation. The distinctive feature of the equation (1) is that, accordingly to [2], no differential substitution exists connecting it with other KdV-type...

Dimension breaking occurs when the solution of a nonlinear partial differential equation (PDE) depending on n independent variables bifurcates to one depending on n + 1. A central hypothesis in the theory of dimension breaking is that a certain operator should have a non-zero purely imaginary eigenvalue. This hypothesis is difficult to verify in general. We present a geometric theory for verify...