A modification of the method of characteristics: a new class of multidimensional partially integrable nonlinear systems

TL;DR

The paper introduces a generalized method of characteristics to reduce multidimensional nonlinear PDEs to lower-dimensional systems, enabling partial integration and potential applications in hydrodynamics.

Contribution

It presents an algorithm that reduces (M+1)-dimensional nonlinear PDEs to M-dimensional first order PDEs, generalizing the method of characteristics for broader classes of systems.

Findings

Reduces system dimensionality by integrating with respect to one variable.

Applicable to a wide class of nonlinear PDEs, including hydrodynamics.

Provides a new approach for partial integrability of complex systems.

Abstract

We represent an algorithm reducing a big class of systems of ($M+1$)-dimensional nonlinear partial differential equations (PDEs) to the systems of $M$-dimensional first order PDEs. Thus, we integrate the original system with respect to only one independent variable reducing its dimensionality by one. For this reason we call such systems partially integrable ones. In particular, if M=1, then $M$-dimensional PDEs become ODEs. Our method may be referred to as a generalization of the method of characteristics. Possible application in hydrodynamics is discussed.