Search results for "Vries equation"
showing 10 items of 10 documents
Complex singularities in KdV solutions
2016
In the small dispersion regime, the KdV solution exhibits rapid oscillations in its spatio-temporal dependence. We show that these oscillations are caused by the presence of complex singularities that approach the real axis. We give a numerical estimate of the asymptotic dynamics of the poles.
Degenerate Riemann theta functions, Fredholm and wronskian representations of the solutions to the KdV equation and the degenerate rational case
2021
International audience; We degenerate the finite gap solutions of the KdV equation from the general formulation given in terms of abelian functions when the gaps tend to points, to get solutions to the KdV equation given in terms of Fredholm determinants and wronskians. For this we establish a link between Riemann theta functions, Fredholm determinants and wronskians. This gives the bridge between the algebro-geometric approach and the Darboux dressing method.We construct also multi-parametric degenerate rational solutions of this equation.
Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations
2013
The formation of singularities in solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1+1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers' equation and for dispersive regularizations as the Korteweg-de Vries equation, the Fourier coefficients indicate as …
A numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions
2012
Abstract We study numerically the small dispersion limit for the Korteweg–de Vries (KdV) equation u t + 6 u u x + ϵ 2 u x x x = 0 for ϵ ≪ 1 and give a quantitative comparison of the numerical solution with various asymptotic formulae for small ϵ in the whole ( x , t ) -plane. The matching of the asymptotic solutions is studied numerically.
Stationary problems for equation of the KdV type and dynamical r-matrices
1995
We study a quite general family of dynamical $r$-matrices for an auxiliary loop algebra ${\cal L}({su(2)})$ related to restricted flows for equations of the KdV type. This underlying $r$-matrix structure allows to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems. As an example, we discuss the Henon-Heiles system and a quartic system of two degrees of freedom in detail.
Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation
2007
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\e^2$, $\e\ll 1$, 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 difference between the KdV and the asymptotic solution decreases as $\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a…
Trivial S-Matrices, Wigner-Von Neumann Resonances and Positon Solutions of the Integrable Nonlinear Evolution Equations
1996
It is well known that the scattering matrix is different from the unit matrix in the case of 1-dimensional Schrodinger operator with smooth rapidly decreasing nonzero potential. This no more true in the case of the slowly decreasing and oscillating potentials for which the absence of scattering is accompanied by the occurrence of the Wigner-von Neumann resonances embedded in the positive absolutely continuous spectrum. Taken as initial conditions in the KdV like integrable partial differential equations these potentials generate interesting family of explicit solutions. Below we will call them positon or multipositon solutions. The interaction of an arbitrary finite number of positons and s…
Some evolution equations arising in physics
1983
In this paper we consider a new series of evolution equations generalizing the Korteweg-deVries (KdV) and Burgers equations, and we report recent advances on these equations together with the physical phenomena where they arise. In particular we consider a generalized Burgers' equation and we sketch a method for solution in series by using the theory of Sobolevskij and Tanabe. Then we study the KdV equation with nonuniformity terms and we describe various physical interpretation of this equation. We consider various particular cases in which varying solitonic solutions exist. Also we sketch a unicity theorem. Finally modified Burgers-KdV equations are considered.
30 years of finite-gap integration theory
2007
The method of finite-gap integration was created to solve the periodic KdV initial problem. Its development during last 30 years, combining the spectral theory of differential and difference operators with periodic coefficients, the algebraic geometry of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems, had a strong impact on the evolution of modern mathematics and theoretical physics. This article explains some of the principal historical points in the creation of this method during the period 1973–1976, and briefly comments on its evolution during the last 30 years.
Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations
2013
We present a numerical study of solutions to the generalized Kadomtsev-Petviashvili equations with critical and supercritical nonlinearity for localized initial data with a single minimum and single maximum. In the cases with blow-up, we use a dynamic rescaling to identify the type of the singularity. We present a discussion of the observed blow-up scenarios.