Related papers: Coupled system description of perturbed KdV equati…
An approximate perturbed direct homotopy reduction method is proposed and applied to two perturbed modified Korteweg-de Vries (mKdV) equations with fourth order dispersion and second order dissipation. The similarity reduction equations are…
Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these…
For the L^2 subcritical and critical (gKdV) equations, Martel proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N…
The matrix 2x2 spectral differential equation of the second order is considered on x in ($-\infty,+\infty$). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second…
We consider a coupled PDE system between the Burgers equation and the KdV equation to model the interactions between `bore'-like structures and wave-like solitons in shallow water. Two derivations of the resulting Burgers-swept KdV system…
A Hamiltonian pair with arbitrary constants is proposed and thus a sort of hereditary operators is resulted. All the corresponding systems of evolution equations possess local bi-Hamiltonian formulation and a special choice of the systems…
All solutions of the Korteweg -- de Vries equation that are bounded on the real line are physically relevant, depending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that…
The method for solving the KdV are considered.
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…
In the example of the Schr\"odinger/KdV equation we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's…
The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation…
We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the…
In this work we continue the description of soliton-like solutions of some slowly varying, subcritical gKdV equations. In this opportunity we describe, almost completely, the allowed behaviors: either the soliton is refracted, or it is…
The paper deals with perturbations of the equation that have a number of conservation laws. When a small term is added to the equation its conserved quantities usually decay at individual rates, a phenomenon known as a selective decay.…
It is shown that, three different Lax operators in the Dym hierarchy, produce three generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component KdV system. The first equation…
We give a self-contained introduction to the relations between Integrable Systems and the Geometry of Riemann Surfaces. We start from a historical introduction to the topic of integrable systems. Afterwards, we study the polynomial…
We show that an interacting double soliton solution to the perturbed mKdV equation is close in $H^2$ to a double soliton following an effective dynamics obtained as Hamilton's equations for the restriction of the mKdV Hamiltonian to the…
Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a…
The matrix 2x2 spectral differential equation of the second order is considered on x in ($-\infty,+\infty$). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second…
We consider the general Degasperis-Procesi model of shallow water out-flows. This fife parametric family of conservation laws contains, in particular, KdV, Camassa-Holm, and Degasperis-Procesi equations. The main result consists of a…