and Applied Analysis 3 by considering a general fractional nonlinear nonhomogeneous partial differential equation with the initial condition of the following form: D α t U (x, t) = L (U (x, t)) + N (U (x, t)) + f (x, t) , α > 0, (13) subject to the initial condition D k 0 U (x, 0) = gk, (k = 0, . . . , n − 1) , D n 0 U (x, 0) = 0, n = [α] , (14) where D t denotes without loss of generality the ...

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discusse...

Solitary wave solutions to the Broer-Kaup equations and approximate long water wave equations are considered challenging by using the rst integral method.The exact solutions obtained during the present investigation are new. This method can be applied to nonintegrable equations as well as to integrable ones.

This paper discusses several algorithmic ways of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two component peakon type dual systems from their two component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group...

The generation of an undular bore in the vicinity of a wave-breaking point is considered for the integrable Kaup–Boussinesq (KB) shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the Gurevich–Pitaevskii type of problem for a generic “cubic” breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear Eule...

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...