The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity $f(u) = \tfrac{1}{2}u^2$ of Korteweg-de Vries and full linear dispersion relation $\Omega(k) \sqrt{k\tanh k}$ uni-directional gravity in suitably scaled variables. This paper proposes analyzes generalization Whitham's to unidirectional nonlinear wave equations consisting general fu...

In this paper we discuss the dispersionless limit of the multicomponent 2D Toda hierarchy. The discrete flows of the hierarchy are used to define charge preserving Lax and Orlov–Schulman operators. This construction allows us to perform two types of dispersionless limits, one type leads to the 0-genus universal Whitham hierarchy while the other leads to a dispersionless hierarchy which contains...

ABSTRACT. The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the differ...

The Korteweg de Vries (KdV) equation with small dispersion is a model for the formation and propagation of dispersive shock waves in one dimension. Dispersive shock waves in KdV are characterized by the appearance of zones of rapid modulated oscillations in the solution of the Cauchy problem with smooth initial data. The modulation in time and space of the amplitudes, the frequencies and the wa...

We prove wave breaking (shock formation) for some Whitham-type equations which include the Burgers--Hilbert equation, fractional Korteweg--de Vries and classical Whitham equation. The result seems to be new equation has been proven independently in (Yang, SIAM J. Math. Anal., 53 (2021), pp. 5756--5802). In all cases we provide simpler proofs than known ones.

We collect a number of facts and conjectures concerning Whitham theory and the renormalization group (RG). Some explicit relations and analogues are indicated in the context of N = 2 susy Yang-Mills (YM).

The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of the damped Benjamin-Ono equation. The structure of the dispersive shock is considered in this method.

The Hunter–Saxton equation and the Gurevich–Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly.

Generating functions for a complete collection of symmetries of the multiphased averaged KdV equation are constructed. The isospectral generating function has a potential form with one of the canonical basis holomorphic differentials as a potential and possesses some remarkable properties at double points of the hyperelliptic Riemann surface. A new representation for the characteristic speeds o...

We review recent work on the study of N = 2 super Yang-Mills theory with gauge group SU(N) from the point of view of the Whitham hierarchy, mainly focusing on three main results: (i) We develop a new recursive method to compute the whole instanton expansion of the low-energy effective prepotential; (ii) We interpret the slow times of the hierarchy as additional couplings and promote them to spu...