Another Complex Bateman Equation

TL;DR

This paper explores a new class of complex covariant field equations that are solvable, possess multiple Lagrangians, and are invariant under linear transformations, linking them to hydrodynamic equations.

Contribution

It introduces a novel class of complex covariant equations with unique properties, expanding understanding of their solutions and invariances.

Findings

Equations can be solved or partially solved via implicit relations.

They have an infinite number of inequivalent Lagrangians.

They are invariant under linear transformations of variables.

Abstract

A further class of complex covariant field equations is investigated. These equations possess several common features: they may be solved, or partially solved in terms of implicit functional relations, they possess an infinite number of inequivalent Lagrangians which vanish on the space of solutions of the equations of motion, they are invariant under linear transformations of the independent variables, and thus are signature-blind and are consequences of first order equations of hydrodynamic type.