Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Superimposed travelling wave solutions for nonlinear diffusion

Authors: Desmond L. Hill and James M. Hill
Journal: Quart. Appl. Math. 51 (1993), 633-641
MSC: Primary 35K55; Secondary 35C05
DOI: https://doi.org/10.1090/qam/1247432
MathSciNet review: MR1247432
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Abstract: For one-dimensional nonlinear diffusion with diffusivity $ D\left( c \right) = {c^{ - 1}}$, a new exact solution is noted which relates to both the well-known travelling wave solution and the source solution. The new solution can have a zero initial condition and admits essentially two distinct forms, one involving the hyperbolic tangent function and the other involving the circular tangent function. The solution involving tanh is physically meaningful and is displayed graphically while that involving the tan function is utilized together with a reciprocal Bäcklund transformation to produce a further new solution, which is physically more interesting than the tan solution, and is also displayed graphically. The basic idea used in this paper is generalized to a high-order nonlinear diffusion equation. For a third-order nonlinear diffusion-like equation, a solution reminiscent of solutions of soliton equations and involving the hyperbolic secant function is obtained and displayed graphically. The solutions investigated here are all characterized by the curious property that individually they are solutions for nonlinear diffusion and, moreover, their ``sum'' is also a bonafide nonlinear diffusion solution so that, for this specific solution, a nonlinear partial differential equation displays a ``limited'' superposition principle.

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DOI: https://doi.org/10.1090/qam/1247432
Article copyright: © Copyright 1993 American Mathematical Society

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