In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider BWBK equations, Whitham–Broer–Kaup (WBK) Boussinesq by setting proper smooth functions that are derived from proposed equation. obtain uniformly coupled traveling wave solutions of considered employing improved system method, subsequently their ...

We present a way to deal with dispersion-dominated "shock-type" transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic regularized by a small amount of dispersion. The analysis is performed by assuming that the dispersive shock transition between two different constant states can be modeled by an expansion fan solution of th...

CONTENTS Introduction 35 Chapter I. Hamiltonian theory of systems of hydrodynamic type 45 § 1. General properties of Poisson brackets 45 §2. Hamiltonian formalism of systems of hydrodynamic type and 55 Riemannian geometry §3. Generalizations: differential-geometric Poisson brackets of higher orders, 66 differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation §4. Rieman...

Fractional Fornberg-Whitham equation has a vast application in physics. There exist various investigations for the above problem by considering the variables and parameters as crisp/exact. In practice, we may not have these parameters exactly but those may be known in some uncertain form. In the present paper, these uncertainties are taken as interval/fuzzy and the authors proposed here a new m...

We survey some topics involving the Whitham equations, concentrating on the role of ψψ∗ (or square eigenfunctions) in averaging and in producing Cauchy kernels and differentials on Riemann surfaces.

In this paper, new travelling wave solutions to the Fornberg-Whitham equation ut − uxxt + ux + uux = uuxxx + 3uxuxx are investigated. They are characterized by two parameters. The expresssions of the periodic and solitary wave solutions are obtained.

A complete set of conditions describing the transition across the dissipationless undular bore (dispersive shock) is derived for nonlinear weakly dispersive conservative systems that generally are not integrable by the inverse spectral transform method. In the absence of integrable structure, we postulate modulation description of the undular bore with the aid of the averaged Whitham equations ...