Linear superposition in the general heavenly equation

TL;DR

This paper demonstrates that under certain conditions, nonlinear effects in the general heavenly equation can balance to allow for nontrivial linear superpositions, revealing new solution structures in complex systems.

Contribution

It shows how specific nonlinear effects can be balanced to enable linear superpositions in the general heavenly equation, a novel insight in nonlinear analysis.

Findings

Existence of implicit shock wave solutions with three arbitrary functions

Linear superpositions can be approximately realized through nonlinear effect balancing

New solutions constructed from restricted arbitrary functions

Abstract

Evidently, the linear superposition principle can not be exactly established as a general principle in the presence of nonlinearity, and, at the first glance, there is no expectation for it to hold even approximately. In this letter, it is shown that the balance of different nonlinear effects describes what linear superpositions may occur in nonlinear systems. The heavenly equations are of significance in several scientific fields, especially in relativity, gravity, field theory, and fluid dynamics. A special type of implicit shock wave solution with three two-dimensional arbitrary functions of the general heavenly equation is revealed. Restrict the two-dimensional arbitrary functions to some types of one-dimensional arbitrary functions, it is found that the nonlinear effects can be balanced such that the "impossible" linear superposition solutions can be nontrivially constituted to new solutions of the general heavenly equation.