Lax Pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2 + 1)-Dimensional Equation

Abstract

We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2 + 1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional CKdV equation. We show that the CKdV equation as well as the (2+1)-dimensional CKdV equation are integrable in the sense that they possess the Painlevé property. Some exact solutions are also constructed.

A Series of Exact Solutions for a New (2+1)-Dimensional Calogero KdV Equation

“…This equation was introduced by Yu and Toda recently [12] by considering extension of the Calogero KdV equation. [13,14] They further showed that equation (30) was integrable in the sense of Painlevé property.…”

Symmetry analysis, optimal classification and dynamical structure of exact soliton solutions of (2+1)-dimensional modified Bogoyavlenskii–Schiff equation

“…where u is a function in x, y, t variables and subscript stand for partial derivatives w.r.t its variables x, y and t, respectively. The initial form of the mBSchiff equation was given in the form of a partial integrodifferential equation [23][24][25][26][27] where…”

Nonlocal symmetry and exact solutions of the (2+1)-dimensional modified Bogoyavlenskii–Schiff equation

“…In the present paper, we focus on investigating the nonlocal symmetry, prolonged system, Bäcklund transformation, CRE solvability, and the exact interaction solutions of the following (2+1)-dimensional modified Bogoyavlenskii-Schiff (mBS) equation [39][40][41][42][43] u…”