Bäcklund transformations of soliton systems from symmetry constraints.

*(English)* Zbl 1010.37048
Coley, Alan (ed.) et al., Bäcklund and Darboux transformations. The geometry of solitons. AARMS-CRM workshop, Halifax, Canada, June 4-9, 1999. With short biographies of Albert Victor Bäcklund and Gaston Darboux. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes. 29, 313-323 (2001).

Summary: Binary symmetry constraints are applied to constructing Bäcklund transformations of soliton systems, both continuous and discrete. Construction of solutions to soliton systems is split into finding solutions to lower-dimensional Liouville integrable systems, which also paves a way for separation of variables and exhibits integrability by quadratures for soliton systems. Illustrative examples are provided for the KdV equation, the AKNS system of nonlinear Schrödinger equations, the Toda lattice, and the Langmuir lattice.

##### MSC:

37K35 | Lie-Bäcklund and other transformations |

35Q58 | Other completely integrable PDE (MSC2000) |

37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws |

39A10 | Additive difference equations |