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Melnikov Analysis for a Singularly Perturbed DSII Equation
Author(s) -
Li Y. Charles
Publication year - 2005
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.0022-2526.2005.01531.x
Subject(s) - homoclinic orbit , mathematics , singular perturbation , perturbation (astronomy) , center manifold , mathematical analysis , heteroclinic orbit , orbit (dynamics) , transformation (genetics) , manifold (fluid mechanics) , physics , bifurcation , nonlinear system , mechanical engineering , biochemistry , chemistry , hopf bifurcation , quantum mechanics , engineering , gene , aerospace engineering
Rigorous Melnikov analysis is accomplished for Davey–Stewartson II equation under singular perturbation. Unstable fiber theorem and center‐stable manifold theorem are established. The fact that the unperturbed homoclinic orbit, obtained via a Darboux transformation, is a classical solution, leads to the conclusion that only local well posedness is necessary for such a Melnikov analysis. The main open issue regarding a proof of the existence of a homoclinic orbit to the perturbed Davey–Stewartson II equation is discussed in the Appendix.