Frontiers in Nonlinear Waves
in honor of Vladimir Zakharov's 70th birthday
March 26–29, 2010
University of Arizona, Tucson, AZ, USA

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Conference Program

Mar 26 Mar 27 Mar 28 Mar 29 Speakers

Monday March 29, 9:50–10:15

Soliton Reflection in Boundary Value Problems and a Nonlinear Method of Images
Gino Biondini
Department of Mathematics, State University of New York at Buffalo, NY
It is well known that each soliton in an exact N-soliton solution of an integrable nonlinear evolution equation (NLEE) is associated to a discrete eigenvalue of the scattering problem via the inverse scattering transform. Recent results, however, have shown that dynamics in initial-boundary value problems (IBVPs) are more complicated than in initial value problems (IVPs). I will characterize IBVPs for both discrete and continuous nonlinear Schroedinger (NLS) systems on a semi-infinite domain with linearizable boundary conditions (BCs), with either zero or non-zero BCs at infinity.

For NLS on the half line, the linearizable BCs are of homogeneous Robin type. In all of approaches to the IBVP, the relation between solitons and discrete eigenvalues existing in the IVP is preserved in the IBVP. Soliton solutions of the NLS equation, however, do not in general satisfy the linearizable BCs. Moreover, numerical solutions of the IBVP clearly show that the solitons are reflected at the boundary — even though the soliton velocity is determined by the discrete eigenvalue, which is time-independent. I will show that the resolution of these apparent paradoxes is that discrete eigenvalues in the IBVP appear in symmetric pairs, and that corresponding relations exist for the associated normalization constants. For each soliton in the physical domain a symmetric counterpart exists, with equal amplitude and opposite velocity, whose presence ensures that the solution satisfies the BCs. The ostensible reflection of the soliton at the boundary corresponds to the interchanging of roles between the “physical” and “mirror” solitons. Similar phenomena arise for the focusing and defocusing NLS and for the the Ablowitz–Ladik lattice. These results represent a nonlinear analogue of the method of images in electrostatics. Here, however, the reflection is accompanied by a position shift, which is a reminder of the nonlinear nature of the problem.

 presentation file (PDF, 1.4 MiB)