New Lax Pairs and Darboux Transformation and Its Application to a Shallow Water Wave Model of Generalized KdV Type

Abstract

New Lax pairs of a shallow water wave model of generalized KdV equation type are presented. According to this Lax pair, we constructed a new spectral problem. By using this spectral problem, we constructed Darboux transformation with the help of a gauge transformation. Applying this Darboux transformation, some new exact solutions including double-soliton solution of the shallow water wave model of generalized KdV equation type are obtained. In order to visually show dynamical behaviors of these double soliton solutions, we plot graphs of profiles of them and discuss their dynamical properties.

Diverse analytical wave solutions and dynamical behaviors of the new (2 + 1)-dimensional Sakovich equation emerging in fluid dynamics

“…Therefore, the exact closed-form solutions of these equations play a fundamental role in unraveling the dynamic and help to understand/comprehend the mechanism of the existent state. A variety of novel approaches have been effectively utilized, developed, and improved by collections of assorted researchers for retrieving the exact solutions of NLEEs where the most important goes back to Lax pair [1], bifurcation method [2,3], extended mapping method [4], the tanh method [5], extended multiple Riccati equations expansion method [6], inverse scattering method [7], extended Jacobian elliptic function expansion method [8,9], Lie symmetry analysis [10][11][12][13][14][15][16], generalized Kudryashov method [17], etc. Although there is no particular universal technique that is applicable to all NLPDEs.…”

“…was introduced by Sakovich in [18]. It was observed that equation (1) which is quadratic in u xx is Painlevé integrable and gives soliton solutions of KdV-type. The KdV equation which is known as Korteweg-de Vries…”

“…Despite the fact that ( 1) is (very close) irrespective to the spectral problem of the KdV equation, it can be stated that the given system of equations ( 2) and ( 3) is a reduction of the equation (1). Due to the representation (4), each function u(x, y,t) which satisfies the system of KdV equations ( 2) and (3) at the same time is also a solution of the new equation (1) as well but converse is not true. Although the wave equation (1)…”

“…Due to the representation (4), each function u(x, y,t) which satisfies the system of KdV equations ( 2) and (3) at the same time is also a solution of the new equation (1) as well but converse is not true. Although the wave equation (1)…”

Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

“…The Lax pairs of (2) were given by Fokas in [40]. New Lax pairs and Darboux transformation of (2) were introduced by Yang and Rui in [43] recently. In [44], by using the bifurcation theory of dynamical system, the existence conditions of different kinds of traveling wave solutions of (2) were presented by Bi.…”