Lax pairs, Painlev\'e properties and exact solutions of the alogero Korteweg-de Vries equation and a new (2+1)-dimensional equation

TL;DR

This paper establishes the integrability of the Calogero Korteweg-de Vries (CKdV) equation and introduces a new (2+1)-dimensional CKdV equation, providing Lax pairs, Painlevé analysis, and exact solutions.

Contribution

It proves the existence of a Lax pair for CKdV, modifies it for a new (2+1)-dimensional equation, and demonstrates their integrability and exact solutions.

Findings

Existence of Lax pair for CKdV

Introduction of a new (2+1)-dimensional CKdV equation

Both equations possess the Painlevé property and are integrable

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 Painlev\'e property. Some exact solutions are also constructed.