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Generalizations of the short pulse equation.

Andrew N W Hone¹, Vladimir Novikov², Jing Ping Wang¹

¹1School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS UK.

Letters in Mathematical Physics

|April 3, 2018

Summary

This study classifies second-order polynomial partial differential equations. These equations generalize the well-known short pulse equation, advancing the field of integrable systems.

Keywords:

Lax pair Recursion operator Short-wave limit Symmetries

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Area of Science:

Mathematical Physics
Nonlinear Dynamics

Background:

The short pulse equation is a significant model in nonlinear optics and fluid dynamics.
Classifying integrable partial differential equations (PDEs) is crucial for understanding their properties and applications.

Purpose of the Study:

To classify integrable scalar polynomial partial differential equations of second order.
To generalize the existing classification to include equations beyond the standard short pulse equation.

Main Methods:

Employing techniques from the theory of integrable systems.
Utilizing algebraic and geometric methods for classifying polynomial PDEs.

Main Results:

A comprehensive classification of a new family of integrable second-order scalar polynomial PDEs.
Identification of generalizations of the short pulse equation within this new class.

Conclusions:

The classification provides a deeper understanding of integrable nonlinear PDEs.
This work expands the toolkit for analyzing complex nonlinear phenomena described by such equations.