We consider the generalized Korteweg-de Vries equation ∂tu + ∂ 3 xu + ∂x(u ) = 0, (t, x) ∈ R2, in the supercritical case p > 5, and we are interested in solutions which converge to a soliton in large time in H. In the subcritical case (p < 5), such solutions are forced to be exactly solitons by variational characterization [1, 19], but no such result exists in the supercritical case. In this pa...

We construct multi-soliton solutions of the n-component vector nonlinear Schr\odinger equation on half-line subject to two classes integrable boundary conditions (BCs): homogeneous Robin BCs and mixed Neumann/Dirichlet BCs. The construction is based approach dressing BCs: soliton are generated in preserving at each step Darboux-dressing process. Under BCs, examples, including boundary-bound sol...

Abstract. 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 lowerdimensional Liouville integrable systems, which also paves a way for separation of variables and exhibits integrability by quadratures for soliton systems. Illustrative ex...

This paper obtains the solitary wave, shock wave as well as singular soliton solutions to the generalized OstrovskyBenjamin-Bona-Mahoney (gO-BBM) equation. The ansatz method is applied to obtain the solutions. Several constraint conditions for the parameters are derived that establish the existence of the soliton solutions.

Some aspects of two-dimensional gravity coupled to matter fields, especially to the Sine-Gordon-model are examined. General properties and boundary conditions of possible soliton-solutions are considered. Analytic soliton-solutions are discovered and the structure of the induced space-time geometry is discussed. These solutions have interesting features and may serve as a starting point for fur...

In this paper many soliton-like solutions with the arbitrary functions of y and t are obtained for the 1+2D Calogero-Bogoyavlenskii-Schif equation by means of some generalized ansatz and symbolic computation. Some new exact solutions are also presented from the obtained soliton-like solutions. Shock wave solution is only a special case.

Based on the factorization of soliton equations into two commuting integrable xand t-constrained flows, we derive N -soliton solutions for mKdV equation via its xand t-constrained flows. It shows that soliton solution for soliton equations can be constructed directly from the constrained flows.

We report an exact bright and dark soliton solution to the nonlinear evolution equation derived by MOSES and WISE (Phys. Rev. Lett. 97, 2006, 073903) for cascaded quadratic media beyond the slowly varying envelope approximations. The integrability aspects of the model are addressed. The travelling wave hypothesis as well as the ansatz method are employed to obtain an exact 1-soliton solution. B...