From fully-nonlinear to semilinear evolution equations: two symmetry-integrable examples

TL;DR

This paper derives two new fully-nonlinear symmetry-integrable evolution equations of third and fifth order, using symmetry analysis and transformations to relate them to known semilinear integrable equations.

Contribution

It introduces two novel fully-nonlinear symmetry-integrable evolution equations and demonstrates their connection to known semilinear equations through advanced transformations.

Findings

Derived two new fully-nonlinear integrable equations of orders 3 and 5

Mapped these equations to known semilinear integrable equations

Explicitly presented related symmetry-integrable quasilinear equations of orders 5 and 7

Abstract

In this paper we derive two examples of fully-nonlinear symmetry-integrable evolution equations with algebraic nonlinearities, namely one class of 3rd-order equations and a 5th-order equation. To achieve this we study the equations' Lie-B\"acklund symmetries and apply multipotentialisations, hodograph transformations and generalised hodograph transformations to map the equations to known semilinear integrable evolution equations. As a result of this, we also obtain interesting symmetry-integrable quasilinear equations of order five and order seven, which we display explicitly.