A Brief Survey on Constructing homoclinic Structures of soliton Equations

To give rigorous mathematical proofs of chaotic behaviors in a given system, it is necessary to identify the homoclinic structures in the system. In this tutorial review, methods for constructing explicit solutions for nonlinear partial differential equations are presented, with more emphasis placed on those utilizing complete integrability associated with soliton equations. As an extended application, homoclinic orbits to spatial uniform plane waves of coupled modified nonlinear Schrodinger equations are obtained via the dressing method. During the procedure, it is necessary to introduce the Lax pair for these coupled equations, as well as its Floquet spectral analysis and corresponding Bloch functions.

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