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March 17 Math Colloquium

Baofeng Feng, University of Texas Rio Grande Valley

KP-Toda hierarchy reduction method for soliton equations

We give a review for the KP-Toda hierarchy reduction method to find multi-soliton and rogue wave solutions to soliton equations which was developed by Kyoto School. The KP- Toda hierarchy reduction method usually starts with the bilinear equations satisfied by the tau functions of the KP-Toda hierarchy, then through a series of reductions such as dimension and symmetry reductions, finally the multi-soliton or rogue wave solution can be derived in either determinant or Pfaffian form. In this talk, I will take the nonlinear Schrodinger (NLS) equation and its vector form as an example to show how we can derive the multi-bright, multi-dark and mixed soliton solutions. If time permits, I will report nonsingular soliton solutions with zero and nonzero boundary conditions to a nonlocal NLS equation with PT-symmetry.

This is a joint work with Mark Ablowitz, Yasuhiro Ohta and Kenichi Maruno.

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