We consider the modulationally stable version of Kaup–Boussinesq system which models propagation nonlinear waves in various physical situations. It is shown that Whitham modulation equations for this model have a new type solutions describe trigonometric shock waves. In Gurevich–Pitaevskii problem evolution an initial discontinuity, these correspond to nonzero wave excitation on one sides disco...

Many methods have been developed by scientists to find solutions for nonlinear problems. In this paper, the general structure of modified trial equation method (MTEM) is introduced, and MTEM used some exact (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK), Kolmogorov-Petrovskii-Piskunov (KPP) equations. Firstly, an algebraic system obtained reducing partial differential (NLPDE) ordinary under tra...

— Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different from the original ones. Binary nonlinearization procedure corresponding to the Bargmann symmetry constraint is carried out for those Lax pairs. The proposed Lax pairs together with adjoint Lax pairs are constrained as a hierarchy of commutative, finite dimensional integrable Hamiltonian systems in the Liouville sense,...

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...

In this paper, we present new Hamiltonian operators for the integrable couplings of Ablowitz–Kaup–Newell–Segur hierarchy and Kaup–Newell hierarchy. The corresponding Hamiltonians allow nontrivial degeneration. Multi-Hamiltonian structures are investigated. involutive property is proven known with respect to two Poisson brackets defined by operators.

In this work, a novel integrable evolution system in the sense of Lax’s scheme associated with mixed spectral Ablowitz-Kaup-Newell-Segur (AKNS) matrix problem is first derived. Then, time dependences scattering data corresponding to AKNS are given inverse analysis. Based on data, reconstruction potentials carried out, and finally analytical solutions four arbitrary functions derived formulated....

The ( 2 + 1 )-dimensional Lax integrable equation is decomposed into solvable ordinary differential equations with the help of known id="M3"> soliton associated Ablowitz-Kaup-Newell-Segur hierarchy. Then, based on finite-order expansion matrix, a hyperelliptic Riemann surface and Abel-Jacobi coordinates...