Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations studied. The fourth order differential equation traveling waves a parameter in front of derivative is reduced to reversible Hamiltonian system defined on two-sheeted four-dimensional space. Solutions which stay one sheet represent smooth but those perform transitions through branchin...

Abstract Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves shallow water. We prove that, under a non-cavitation condition on initial data, these are well-posed time scale order <mml:math xmlns:mml="http://www.w3.org/1998/Math...

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and good approximation the incompressible Euler equations. In subsequent papers (Dinvay, 2019; al., 2019) initial-value studied well-posedness classical Sobolev spaces proved. He...

We describe an expedient way to derive the CIV-DV prepotential in power series expansion in Si. This is based on integrations of equations for its derivatives ∂F/∂Tm with respect to additional (Whitham) moduli Tm. For illustrative purposes, we calculate explicitly the leading terms of the expansion and explicitly check some components of the WDVV equations to the leading order. Extension to any...

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...

We review new aspects of integrable systems discovered recently in N = 2 supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten curves for classical gauge groups, (ii) its application to contact terms in the u-plane integral of topologically twisted theories, and (iii) a connection be...

In the present paper scalar macroscopic models for traffic and pedestrian flows are coupled and the resulting system is investigated numerically. For the traffic flow the classical Lighthill-Whitham model on a network of roads and for the pedestrian flow the Hughes model are used. These models are coupled via terms in the fundamental diagrams modeling an influence of the traffic and pedestrian ...

We prove the existence of a global bifurcation branch of 2π-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class C, α < 1 2 . Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spec...

We consider the initial value problem for generalized Fornberg-Whitham equation, which consists of p-order nonlinear advection term and nonlocal dispersion term. In case p=2, we study sufficient conditions blow-up solutions some relations to Burgers equation. Also, p?5, show global existence with small data.

In this paper, we prove that the Cauchy problem for Fornberg–Whitham equation is not locally well-posed in Bp,rs(R) with (s,p,r)∈(1,1+1p)×[2,∞)×[1,∞] or (s,p,r)∈{1}×[2,∞)×[1,2] by showing norm inflation phenomena of solution some special initial data.