Variable coefficient Davey-Stewartson system with a Kac-Moody-Virasoro symmetry algebra

TL;DR

This paper analyzes the symmetry properties of the variable coefficient Davey-Stewartson system, identifying conditions under which it is equivalent to the standard DS system and exploring the implications for integrability and solutions.

Contribution

It characterizes the symmetry algebra of the vcDS system, establishes conditions for equivalence to the DS system, and derives explicit symmetry groups and solutions.

Findings

Symmetry algebra of vcDS is isomorphic to DS under specific coefficient conditions.

Integrable subsystems are identified using the equivalence group.

Lump solutions exist for the vcDS system when it is equivalent to DS.

Abstract

We study the symmetry group properties of the variable coefficient Davey-Stewartson (vcDS) system. The Lie point symmetry algebra with a Kac-Moody-Virasoro (KMV) structure is shown to be isomorphic to that of the usual (constant coefficient) DS system if and only if the coefficients satisfy some conditions. These conditions turn out to coincide with those for the vcDS system to be transformable to the DS system by a point transformation. The equivalence group of the vcDS system is applied to pick out the integrable subsystems from a class of non-integrable ones. Additionally, the full symmetry group of the DS system is derived explicitly without exponentiating its symmetry algebra. Lump solutions (rationally localized in all directions in $\mathbb{R}^2$) introduced by Ozawa for the DS system is shown to hold even for the vcDS system precisely when the system belongs to the integrable class, i.e. equivalent to the DS system. These solutions can be used for establishing exact blow-up solutions in finite time in the space $L^2(\mathbb{R}^2)$ in the focusing case.