Vol 197, No 1 (2018)

Nonlinear generalizations of integrable equations in one dimension, such as the Korteweg–de Vries and Boussinesq equations with p-power nonlinearities, arise in many physical applications and are interesting from the analytic standpoint because of their critical behavior. We study analogous nonlinear p-power generalizations of the integrable Kadomtsev–Petviashvili and Boussinesq equations in two dimensions. For all p ≠ 0, we present a Hamiltonian formulation of these two generalized equations. We derive all Lie symmetries including those that exist for special powers p ≠ 0. We use Noether’s theorem to obtain conservation laws arising from the variational Lie symmetries. Finally, we obtain explicit line soliton solutions for all powers p > 0 and discuss some of their properties.

Our aim is to develop the inverse scattering transform for multicomponent generalizations of nonlocal reductions of the nonlinear Schrödinger (NLS) equation with \(\mathcal{PT}\) symmetry related to symmetric spaces. This includes the spectral properties of the associated Lax operator, the Jost function, the scattering matrix, the minimum set of scattering data, and the fundamental analytic solutions. As main examples, we use theManakov vector Schrödinger equation (related to A.III-symmetric spaces) and the multicomponent NLS (MNLS) equations of Kulish–Sklyanin type (related to BD.I-symmetric spaces). Furthermore, we obtain one- and two-soliton solutions using an appropriate modification of the Zakharov–Shabat dressing method. We show that the MNLS equations of these types admit both regular and singular soliton configurations. Finally, we present different examples of one- and two-soliton solutions for both types of models, subject to different reductions.

By the method of dressing on a torus, we obtain and study solutions of the Landau–Lifshitz equation, which describe solitons in the stripe domain structure of the easy-axis ferromagnet. A specific feature of these solitons is that they are directly related to the domain structure: they induce translations and local oscillations of the domains. We find integrals of motion stabilizing the solitons on the background of the structure.

We study the Hamiltonian geometry of systems of hydrodynamic type that are equivalent to the associativity equations in the case of three primary fields and obtain the complete classification of the associativity equations with respect to the existence of a first-order Dubrovin–Novikov Hamiltonian structure.

We study the behavior of the soliton that encounters a barrier with dissipation while moving in a nondissipative medium. We use the Korteweg–de Vries–Burgers equation to model this situation. The modeling includes the case of a finite dissipative layer similar to a wave passing through air–glass–air and also a wave passing from a nondissipative layer into a dissipative layer (similar to light passing from air to water). The dissipation predictably reduces the soliton amplitude/velocity. Other effects also occur in the case of a finite barrier in the soliton path: after the wave leaves the dissipative barrier, it retains the soliton form, but a reflection wave arises as small and quasiharmonic oscillations (a breather). The breather propagates faster than the soliton passing through the barrier.