Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation

Details Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation

2002-09-22

arxiv.org/abs/nlin/0209045v3

Physica D 176 (2003) 44-66

Nonlinear Sciences

Pattern Formation and Solitons

32 pages, no figures, elsart.cls

Scientific paper

10.1016/S0167-2789(02)00708-X

Using Painlev\'e analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass $\wp$ function. Although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic Swift-Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far.

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