| In this paper, by using Painleve analysis, the Hirota multi-linear method and a di-rect ansatz technique, the analytic solutions of one-dimensional nonlinear cubic-quintic complex Swift-Hohenberg equation with a dispersion and the exact solutions for do-main walls(also known as "front" and "kinks") in nonlinear coupled cubic-quintic com-plex Ginzburg-Landau equations are investigated, respectively. We hope that these re-sults could provide a profound value for the studies of optical and slowly varying wave packets in nonlinear dissipative media. This paper is divided into four chapters:The first chapter is the introduction. It briefly introduces the application back-ground of the Swift-Hohenberg equation and Ginzburg-Landau equation and summa-rizes the contents of this paper.In second chapter of this paper, analytic solutions of the one-dimensional nonlin-ear cubic-quintic complex Swift-Hohenberg equation with a dispersion are obtained by using Painleve analysis, the Hirota multi-linear method and a direct ansatz technique. We can find that numbers of exact solutions exist to every equation and provide that the coefficients are restricted by certain relations. The set of solutions consist of particular types of solitary wave solutions, hole(dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions. In fact, these multi-parameter families of solutions can be acted as a seeding set of solutions which could be very useful in optical studies.In third chapter of this paper, by using a modified Hirota bilinear operator, a novel factorization procedure and numerical simulations, we have found exact solutions for domain walls(also known as "front" and "kinks") in nonlinear coupled cubic-quintic complex Ginzburg-Landau equations. We hope that these solutions could be used for the studies of slowly varying wave packets in nonlinear dissipative media.In fourth chapter of this paper, we summarize those results in this paper and put forward to some open problems. |
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