SUPERIMPOSED TRAVELING-WAVE SOLUTIONS FOR NONLINEAR DIFFUSION

For one-dimensional nonlinear diffusion with diffusivity D(c) = c(-1), a new exact solution is noted which relates to both the well-known tr avelling wave solution and the source solution. The new solution can h ave a zero initial condition and admits essentially two distinct forms , one involving the hyperbolic tangent function and the other involvin g the circular tangent function. The solution involving tanh is physic ally meaningful and is displayed graphically while that involving the tan function is utilized together with a reciprocal Backlund transform ation to produce a further new solution, which is physically more inte resting 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 equati on, a solution reminiscent of solutions of soliton equations and invol ving the hyperbolic secant function is obtained and displayed graphica lly. The solutions investigated here are all characterized by the curi ous property that individually they are solutions for nonlinear diffus ion and, moreover, their ''sum'' is also a bonafide nonlinear diffusio n solution so that, for this specific solution, a nonlinear partial di fferential equation displays a ''limited'' superposition principle.