## Breather solutions of $$N$$-wave equations.(English)Zbl 1123.35342

Mladenov, Ivaïlo (ed.) et al., Proceedings of the 8th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 9–14, 2006. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-37-0/pbk). 184-200 (2007).
Summary: We consider $$N$$-wave type equations related to symplectic and orthogonal algebras. We obtain their soliton solutions in the case when two different $$\mathbb{Z}_2$$ reductions (or equivalently one $$\mathbb{Z}_2\times \mathbb{Z}_2$$-reduction) are imposed. For that purpose we apply a particular case of an auto-Bäcklund transformation – the Zakharov-Shabat dressing method. The corresponding dressing factor is consistent with the $$\mathbb{Z}_2\times\mathbb{Z}_2$$-reduction. These soliton solutions represent $$N$$-wave breather-like solitons. The discrete eigenvalues of the Lax operators connected with these solitons form “quadruplets” of points which are symmetrically situated with respect to the coordinate axes.
For the entire collection see [Zbl 1108.53003].

### MSC:

 35Q51 Soliton equations 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds