In this short note we discuss a new formula for solving the nonlocal $\overline{\partial}$-problem, and application to Manakov--Zakharov dressing method. We then explicitly apply complex (2+1)D Kadomtsev--Petviashvili equation completely integrable generalization of Kaup--Broer (or Kaup--Boussinesq) system. will also how real (1+1)D solutions are expressed using formalism. It is simple express ...

The interactions between soliton solutions of integrable models are usually considered to be completely elastic. That is to say, the amplitude, velocity and wave shape of solitons are not changed after nonlinear interaction [1]. However, for some specific solutions of some (2+1)-dimensional models, the interactions among solitonic excitations are not completely elastic since their shapes are ch...

Based on two higher-dimensional extensions of Lie algebras, three kinds of specific Lie algebras are introduced. Upon constructing proper loop algebras, six isospectral matrix spectral problems are presented and they yield nonlinear integrable couplings of the AblowitzKaup-Newell-Segur hierarchy, the Broer-Kaup hierarchy and the Kaup-Newell hierarchy. Especially, the reduced cases of the result...

By applying a special Bäcklund transformation, a general variable separation solution for the (2+1)-dimensional higher-order Broer-Kaup system is derived. In addition to some types of the usual localized excitations, such as dromions, lumps, ring solitons, oscillated dromions and breathers, soliton structures can be easily constructed by selecting arbitrary functions appropriately. A new class ...

Abstract We analyze the Broer-Kaup system posed on half-line by using unified transform method, also known as Fokas method. derive formal representation of solution for in terms matrix Riemann-Hilbert problem formulated complex plane spectral parameter. The jump is uniquely defined functions that satisfy a certain relation, called global relation involving initial and boundary values. Furthermo...