Related papers: On the Whitham Equations for the Defocusing Comple…
The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated $N$-phase nonlinear wave solutions to the focusing nonlinear Schr\"odinger (fNLS) equation, and b) the Riemann-Hilbert Problem approach to…
We get the leading term of the Gurevich-Pitaevskii special solution to the KdV equation in the oscillation zone without using averaging methods.
The long time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third order dispersion is considered by use of Whitham averaging. Under modest…
Generalized solitary waves with exponentially small non-decaying far field oscillations have been studied in a range of singularly-perturbed differential equations, including higher-order Korteweg-de Vries (KdV) equations. Many of these…
We present a fundamental solution to an initial value problem for the KdV-Whitham system in an explicit integral form. Monotonically decreasing initial data with finite number of breaking points are considered. Generating function for the…
We study the asymptotics of the complex modified Korteweg-de Vries equation $\partial_t u + \partial_x^3 u = -|u|^2 \partial_x u$, which can be used to model vortex filament dynamics. In the real-valued case, it is known that solutions with…
We consider a generalization of the mKdV model of shallow water out-flows. This generalization is a family of equations with nonlinear dispersion terms containing, in particular, KdV, mKdV, Benjamin-Bona-Mahony, Camassa-Holm, and…
We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of…
A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and…
In dispersive media, hydrodynamic singularities are resolved by coherent wavetrains known as dispersive shock waves (DSWs). Only dynamically expanding, temporal DSWs are possible in one-dimensional media. The additional degree of freedom…
We consider propagation of instability fronts in conservative nonlinear wave systems by the Whitham method. Whitham modulation equations for periodic solutions of the generalized Klein-Gordon equation are solved in the limit of…
We use the Gurevich-Pitaevskii approach based on the Whitham averaging method for studying the formation of dispersive shock waves in an intense light pulse propagating through a saturable nonlinear medium. Although the Whitham modulation…
The long-time asymptotic solution of the Korteweg-de Vries equation for general, step-like initial data is analyzed. Each sub-step in well-separated, multi-step data forms its own single dispersive shock wave (DSW); at intermediate times…
We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons…
We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive…
We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation…
The propagation of the dispersive shock waves (DSWs) is investigated in the cylindrical Gardner (cG) equation, which is obtained by employing a similarity reduction to the two space one time (2+1) dimensional Gardner-Kadomtsev-Petviashvili…
I consider the problem of weakly nonlinear stability of three-dimensional parity-invariant magnetohydrodynamic systems to perturbations, involving large scales. I assume that the MHD state, the stability of which I investigate, does not…
We prove that the modulational instability criterion of the formal Whitham modulation theory agrees with the spectral stability of long wavelength perturbations of periodic travelling wave solutions to the generalized Whitham equation. We…
Recently proposed nonholonomic deformation of the KdV equation is solved through inverse scattering method by constructing AKNS-type Lax pair. Exact and explicit N-soliton solutions are found for the basic field and the deforming function…