Related papers: Higher order terms in multiscale expansions: a lin…
The long-time evolution of the KdV-type solitons propagating in ferromagnetic materials is considered trough a multi-time formalism, it is governed by all equations of the KdV Hierarchy. The scaling coefficients of the higher order time…
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher order effects. Although this equation has only one conservation law, exact…
We generalize the non-linear one-dimensional equation of a fluid layer for any depth and length as an infinite order differential equation for the steady waves. This equation can be written as a q-differential one, with its general solution…
In this paper, we study a compound Korteweg-de Vries-Burgers equation with a higher-order nonlinearity. A class of solitary wave solutions is obtained by means of a series expansion.
A new type of wave-mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a…
Fractional calculus of variation plays an important role to formulate the non-conservative physical problems. In this paper we use semi-inverse method and fractional variational principle to formulate the fractional order generalized…
The numerical simulation of nonlinear dispersive waves is a central research topic of many investigations in the nonlinear wave community. Simple and robust solvers are needed for numerical studies of water waves as well. The main…
A complete classification of low-order conservation laws is obtained for time-dependent generalized Korteweg-de Vries equations. Through the Hamiltonian structure of these equations, a corresponding classification of Hamiltonian symmetries…
(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the…
Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction.
There is a well explored relationship between quantum mechanical scattering from a potential and the Korteweg-de Vries (KdV) equation of fluid dynamics: if the potential is 'evolved' according to the KdV equation then it will have the same…
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…
The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such…
The Korteweg-de Vries (KdV) equation is of fundamental importance in a wide range of subjects with generalization to multi-component systems relevant for multi-species fluids and cold atomic mixtures. We present a general framework in which…
We study extreme wave formation for the Korteweg-de Vries equation on the torus with random initial data of average size $\epsilon$. We establish a large deviations principle for the supremum of the solution over arbitrarily long polynomial…
We consider the extended Korteweg-de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves. In the slow time formulation this equation generates fast propagating resonant radiation due to the non-convexity of…
The semi-classical Korteweg-deVries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis Whitham theory is constructed to higher order. This allows the order one…
We study here the water-waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known, that for such regimes, a generalization of the KdV…
The Korteweg-de Vries (KdV) equation is known as a universal equation describing various long waves in dispersive systems. In this article, we prove that in a certain scaling regime, a large class of rough solutions to the Boussinesq…
It is shown that equations of the Korteweg-de Vries hierarchy and their conservation laws can be expressed via the whole powers of an integro-differential operator and functions provided by them.