Integral operator approach over octonions to solution of nonlinear PDE

TL;DR

This paper introduces an octonion-based non-commutative integration method for solving nonlinear PDEs, emphasizing symmetry properties and providing new formulas, theorems, and applications in hydrodynamics.

Contribution

It develops a novel octonion-based framework for integrating nonlinear PDEs, including formulas, theorems, and applications not previously explored.

Findings

Formulas for commutators of integral and differential operators

Theorems for solving nonlinear PDEs using octonions

Applications to hydrodynamics and other PDEs

Abstract

Integration of nonlinear partial differential equations with the help of the non-commutative integration over octonions is studied. An apparatus permitting to take into account symmetry properties of PDOs is developed. For this purpose formulas for calculations of commutators of integral and partial differential operators are deduced. Transformations of partial differential operators and solutions of partial differential equations are investigated. Theorems providing solutions of nonlinear PDEs are proved. Examples are given. Applications to PDEs of hydrodynamics and other types PDEs are described.