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 modelled by an expansion fan solution o...

Who would be at your ideal conference? I’d love to be marooned on a desert island (with whiteboards, or at least a smooth sandy beach to scribble on) with Michael Lynch, Ralph Greenspan, Norbert Perrimon, John Mattick, Michael Wade, Thomas Whitham, Sara Via and Allen Moore. Perrimon, Mattick and Greenspan each have their own unique perspectives on how complex gene networks might work, while Whi...

We extend a previous result, namely we show that the solution of the Whitham equations is asymptotically self-similar for generic monotone polynomial initial data with smooth perturbation.

Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the SU(N+1) N = 2 SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the A (2) 2N affine Toda Toda system. Our construction, too, uses fractional powers of the superpotential W (x) that characterizes the curve. We a...

We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling waves are modulationally unstable if their wave number is greater than a critical wave number predicting a Benjamin-Feir type instability and the result qual...

We survey some topics involving the Whitham equations, concentrating on the role of ψψ∗ (or square eigenfunctions) in averaging. Some connections to symplectic geometry and SeibergWitten theory are indicated.

Abstract We derive the Whitham equations from water waves in shallow regime using two different methods, thus obtaining a direct and rigorous link between these models. The first one is based on construction of approximate Riemann invariants for Whitham–Boussinesq system adapted to unidirectional waves. second generalisation Birkhoff’s normal form algorithm almost smooth Hamiltonians bidirectio...

We study the Seiberg-Witten-Whitham equations in the strong coupling regime of the N = 2 super Yang-Mills theory in the vicinity of the maximal singularities. In the case of SU(2) the Seiberg-Witten-Whitham equations fix completely the strong coupling expansion. For higher rank SU(N) they provide a set of non-trivial constraints on the form of this expansion. As an example, we study the off-dia...

We prove that the modulational instability criterion of formal Whitham modulation theory agrees with spectral stability long-wavelength perturbations periodic traveling wave solutions to generalized equation. use standard WKB procedure derive a quasilinear system three equations, written in terms mass, momentum, and number solution. same quantities as parameters rigorous perturbation linearized...