|XXXIX Workshop on Geometric Methods in Physics||27.06-3.07.2021|
|IX School on Geometry and Physics||21-25.06.2021|
Matrix solitons: tropical limit and entwining Yang-Baxter maps
Whereas solitons of scalar integrable partial differential or difference equations interact trivially (except for "phase shifts"), in case of a matrix generalization solitons carry polarizations that change while they meet. A two-soliton solution in this way determines a map of the two-fold direct product of the space of polarizations. In some cases, like matrix KdV and NLS equation, this map has been found to satisfy the (quantum) Yang-Baxter equation. In these lectures, which are based on recent joint work with Aristophanes Dimakis, we explain why and in which sense the Yang-Baxter property should be expected.
Lecture 1: Generating exact solutions via binary Darboux transformations. Examples: matrix KP and matrix 2D-Toda lattice equations. "Tropical limit" of matrix soliton solutions.
Lecture 2: The matrix KP Yang-Baxter map and its Lax system. Further consequences of the Lax system: entwining Yang-Baxter maps.
Lecture 3: Realization of entwining Yang-Baxter maps in matrix KP and 2D-Toda lattice.
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