Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Abstract

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations 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 equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to `ordinary’ partial differential equations, 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−τ), where τ>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. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

Families of Solutions of Multitemporal Nonlinear Schrödinger PDE

“…The exact solutions of multitime NLSE (with oblique derivative) are closely related to the orbits of the direction vector field h. Our techniques for finding these solutions started from this important idea applied to PDEs that contain directional derivative. We follow a different route than those in the papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and carry out the calculations to obtain significative exact solutions of multitime NLSE. This computational paper and the obtained results show that our methods are simple, efficient, straightforward and powerful.…”

“…Let a ∈ R be an arbitrary constant. We choose k = 2a 2 − c. The Equation ( 8) becomes the Equation (15). For this equation we found two solutions of it that depend on one parameter, namely the functions defined by the formulas (16).…”

“…The paper [13] refers to solving the time-dependent Schrödinger equation via Laplace transform on t. A single-time Schrödinger equation in Riemannian setting is studied in the paper [14]. The paper [15] comes closest to our style by offering "methods for constructing complex solutions of nonlinear PDEs using simpler solutions".…”

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

“…The study [159] constructs exact solutions of linear heat and wave equations with proportional delay using the method of separation of variables. Many nonlinear reactiondiffusion PDEs with proportional delay that admit reductions and exact solutions with additive, multiplicative, generalized, and functional separation of variables are described in [160][161][162]. Approximate analytical methods for solving some linear and nonlinear PDEs with proportional delays are considered in [163,164].…”

Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

“…In [83][84][85], the problems of unique solvability and smoothness of linear boundary value problems for elliptic PDEs with dilation or contraction of arguments in higher derivatives are considered. Analytical methods for solving some linear and nonlinear PDEs with proportional delays are discussed in [86][87][88]. In [89], a finite-difference scheme for the numerical integration of firstorder PDEs with constant delay in t and proportional delay in x is constructed.…”