Solitary wave and other solutions for nonlinear heat equations. (English) Zbl 1116.35035

The authors describe a class of nonlinear heat equations with a polynomial nonlinearity for which explicit solutions can be found by a certain ansatz. The method allows to find infinitely many solutions in terms of e.g. Jacobi elliptic functions or (in the case of Fisher’s equation) the Weierstrass \(\wp\)-function. This includes solutions which had already been found by symmetry methods but also some new exact solutions. Vice versa, the authors describe a modification of Fisher’s equation which possesses solitary wave solutions of any given propagation velocity.


35C05 Solutions to PDEs in closed form
35K55 Nonlinear parabolic equations
35Q51 Soliton equations
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