Homoclinic orbits for a perturbed nonlinear Schrödinger equation

In recent years, there have been extensive studies on the existence of homoclinic orbits for nearly integrable Hamiltonian PDEs, which are closely related to chaos. In this work, we consider a perturbed nonlinear Schrodinger equation for u even and periodic in x. The diffusion ieuxx is an unbounded perturbation term. When the diffusion is replaced by its bounded Fourier truncation, Li, McLaughlin, Shatah, and Wiggins [26] proved the existence of homoclinic orbits for the perturbed equation. The method was based on invariant manifolds, foliations, and Melnikov analysis. The unboundedness of the diffusion prevents the equation from being solved for t < 0 for general initial values and destroys some geometric structures, however. We overcome these difficulties and prove the existence of homoclinic orbits for the diffusively perturbed NLS. © 2000 John Wiley & Sons, Inc.

[1]  Y. Li.  Smale Horseshoes and Symbolic Dynamics in Perturbed Nonlinear Schrödinger Equations , 1999 .

[2]  D. McLaughlin,et al.  Homoclinic orbits and chaos in discretized perturbed NLS systems: Part I. Homoclinic orbits , 1997 .

[3]  Stephen Wiggins,et al.  Homoclinic orbits and chaos in discretized perturbed NLS systems: Part II. Symbolic dynamics , 1997 .

[4]  Jalal Shatah,et al.  PERSISTENT HOMOCLINIC ORBITS FOR A PERTURBED NONLINEAR SCHRODINGER EQUATION , 1996 .

[5]  David W. McLaughlin,et al.  Whiskered Tori for Integrable Pde’s: Chaotic Behavior in Near Integrable Pde’s , 1995 .

[6]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[7]  Xiao-Biao Lin,et al.  Homoclinic Bifurcations with Weakly Expanding Center Manifolds , 1996 .

[8]  J. Shatah,et al.  Homoclinic orbits for the perturbed sine‐Gordon equation , 2000 .

[9]  J. Pöschel,et al.  Inverse spectral theory , 1986 .

[10]  S. Wiggins,et al.  Homoclinic Orbits in a Four Dimensional Model of a Perturbed NLS Equation: A Geometric Singular Perturbation Study , 1996 .

[11]  S. Wiggins,et al.  Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations , 1997 .

[12]  David W. McLaughlin,et al.  Morse and Melnikov functions for NLS Pde's , 1994 .

[13]  W. Strauss.  Nonlinear invariant wave equations , 1978 .

[14]  S. Wiggins,et al.  Orbits homoclinic to resonances: the Hamiltonian case , 1993 .

[15]  G. Haller,et al.  Homoclinic jumping in the perturbed nonlinear Schrödinger equation , 1999 .

[16]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[17]  G. Kovačič,et al.  Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation , 1992 .

[18]  D. McLaughlin.  Whiskered tori for NLS equations , 1993 .

[19]  Jerrold E. Marsden,et al.  A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam , 1981 .

[20]  Stephen Wiggins,et al.  Global Bifurcations and Chaos , 1988 .

[21]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[22]  Peter W. Bates,et al.  Persistence of Overflowing Manifolds for Semiflow , 1999 .

[23]  Stephen Wiggins.  Global Bifurcations and Chaos: Analytical Methods , 1988 .

[24]  G. Haller,et al.  Multi-Dimensional Homoclinic Jumping and the Discretized NLS Equation , 1998 .

[25]  G. Kovačič.  Singular perturbation theory for homoclinic orbits in a class of near-integrable dissipative systems , 1995 .

[26]  David W. McLaughlin,et al.  Geometry of the modulational instability III. Homoclinic orbits for the periodic sine-Gordon equation , 1990 .