Construction of complex solutions to nonlinear partial differential equations using simpler solutions

Abstract

The paper describes a number of simple but quite effective methods for constructing exact solutions of PDEs, that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat/diffusion equations, wave type equations, Klein--Gordon type equations, hydrodynamic boundary layer equations, Navier--Stokes equations, and some other PDEs. Apart from exact solutions to `ordinary' PDEs, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, $u=u(x,t)$, these equations contain the same function at a past time, $w=u(x,t-\tau)$, where $\tau>0$ is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which in addition to the unknown $u=u(x,t)$, also contain the same functions with dilated or contracted arguments, $w=u(px,qt)$, where $p$ and $q$ are scaling parameters.