Finite-gap elliptic solutions of the KdV equation

A method is proposed for constructing finite-gap elliptic inx or/and int solutions of the Korteweg-de Vries equation. Dynamics of poles for two-gap elliptic solutions of the KdV equation are considered. Numerous examples of new elliptic solutions of the KdV equation are given.Dedicated to the memory of J.-L. Verdier     

Korteweg-de Vries hierarchy as an asymptotic limit of the Boussinesq system

For the model of surface waves, we perform an asymptotic analysis with respect to a small parameter ε for large times where corrections to the approximation described by the Korteweg-de Vries equation must be taken into account. We reveal the appearance of the Korteweg-de Vries hierarchy, which ensures the construction of an asymptotic representation up to the times t ≈ ε⁻², where the Korteweg-de Vries approximation becomes inapplicable. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 294–304, February, 2008.     

Disturbed critical surface waves in a channel of arbitrary cross section

Long waves in a current of an inviscid fluid of constant density flowing through a channel of arbitrary cross section under disturbances of pressure distribution on free surface and obstructors on the wall of the channel are considered. The first order asymptotic approximation of the elevation of the free surface satisfies a forced Korteweg-de Vries equation when the current is near its critical state. To determine the coefficients of the forced Korteweg-de Vries equation, we need to solve a linear Neumann problem of an elliptic partial differential equation, of which analytical solutions are found for constant current and rectangular or triangular cross section of the channel. It is proved that the forced Korteweg-de Vries equation has at least two solutions when the current is supercritical and the parameter is greater than a critical value _c >0. It is also proved that there do not exist solitary waves in a current exactly at its critical state. Numerical solutions of steady state are obtained for both supercritical and subcritical currents.     

Bäcklund Algebra and the KdV Hierarchy

This article presents the Korteweg-de Vries hierarchy in the framework of the Lie algebra of B?cklund transformations and points to the problems raised by its vanishing residues characterization. This paper is in final form and no version of it will be submitted for publication elsewhere. Leonardo da Vinci Lecture held on April 26, 2004 Received: February 2005     

Symmetries and Their Lie Algebra of a Variable Coefficient Korteweg-de Vries Hierarchy

Isospectral and non-isospectral hierarchies related to a variable coefficient Painlev′e integrable Korteweg-de Vries(Kd V for short) equation are derived. The hierarchies share a formal recursion operator which is not a rigorous recursion operator and contains t explicitly. By the hereditary strong symmetry property of the formal recursion operator, the authors construct two sets of symmetries and their Lie algebra for the isospectral variable coefficient Korteweg-de Vries(vc Kd V for short) hierarchy.     

Two-gap elliptic solutions to integrable nonlinear equations

We study spectral surfaces associated with elliptic two-gap solutions to the nonlinear Schrödinger equation (NLS), the Korteweg-de Vries equation (KdV), and the sine-Gordon equation (SG). It is shown that elliptic solutions to the NLS and SG equations, as well as solutions to the KdV equation elliptic with respect tot, can be assigned to any hyperelliptic surface of genus 2 that forms a covering over an elliptic surface.     

Solution of the general KdV equation in the class of step functions

In this work, we deduce laws of evolution of the scattering data for the Sturm—Liouville operator with a potential that is a solution of the general Korteweg-de Vries equation and general Korteweg-de Vries equation with a source in the class of step functions. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 174–199.     

Vanishing residue characterization of the sine-Gordon hierarchy

The sine(hyperbolic)-Gordon hierarchy is shown to be the extension of the modified Korteweg-de Vries (MKdV) hierarchy in the integrodifferential algebra extending the standard differential algebra by means of one antiderivative. The characterization by vanishing residues of the MKdV hierarchy yields the same characterization of the sine(hyperbolic)-Gordon hierarchy in the integrodifferential algebra.     

On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KDV potentials

We characterize the spectrum of one-dimensional Schrödinger operatorsH=?d ² /dx ² +V inL ²(?dx) with quasi-periodic complex-valued algebro-geometric potentialsV, i.e., potentialsV which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy, associated with nonsingular hyperelliptic curves. The spectrum ofH coincides with the conditional stability set ofH and can be described explicitly in terms of the mean value of the inverse of the diagonal Green’s function ofH. As a result, the spectrum ofH consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and are discussed as well. These results extend to theL ^p (?dx) forp∈[1, ∞).     

On a class of initial-boundary value problems for equations of Korteweg-de Vries type

We consider a special class of initial-boundary value problems on the positive halfline x > 0 for the Korteweg-de Vries equation and its generalizations. For this class, we prove theorems on the nonexistence of global solutions for t > 0.     

On the nonexistence of global solutions of some initial-boundary value problems for the Korteweg-de Vries equation

We consider the problem of finite-time blow-up of solutions of a class of initial-boundary value problems for the Korteweg-de Vries equation. By using the method of optimal test functions corresponding to the boundary conditions, we obtain blow-up conditions for local (with respect to t > 0) solutions and estimate the blow-up time.     

Method of Immersion and Its Parametric Realization for the Korteweg–De Vries Equation

By using the method of immersion (imbedding) proposed in the author's previous works, we describe the space S of initial conditions of the Cauchy problem for the general differential Korteweg–de Vries equation. The space S is called a stationary soliton Korteweg–de Vries manifold because "stationary projections" of solitons fall into the space S. In addition, we introduce the notion of a space of Sturm–Liouville operators over a soliton Korteweg–de Vries manifold. For real functions and parameters, we formulate the spectral theorem for a commutative Lax pair over a real stationary soliton Korteweg–de Vries manifold.     

Analytical solutions of KdV equation with relaxation effect of inhomogeneous medium

KdV (Korteweg-de Vries) equation with relaxation effect of inhomogeneous medium with time changing can be employed in many different physical fields. In this paper, some new analytical solutions of the equation are obtained, which may be very useful in numerical simulation, by using of the truncated expansion and Jacobi elliptic function expansion methods.     

Integrable and nonintegrable cases of the Lax equations with a source

The Korteweg-de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that, depending on the choice of the basis of the eigenfunctions, we have the following three possibilities: (1) evolution equations for the scattering data are nonintegrable; (2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg-de Vries equation with a source at somet>t _o leaves the considered class of functions decreasing rapidly enough asx±; (3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg-de vries equation with a source exists at allt>t _o. All these possibilities are widespread and occur in other Lax equations with a source.Bogoliubov Theoretical Laboratory, Joint Institute for Nuclear Research 141980 Dubna, Moscow Region, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 471–477, June 1994.     

Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non dispersion.     

A new derivation of Painlevé hierarchies

We give a new derivation of two Painlevé hierarchies. This is done by extending the accelerating-wave reductions of the Korteweg-de Vries and dispersive water wave equations to their respective hierarchies. We also consider the extension of this reduction of Burgers equation to the Burgers hierarchy.     

Global Well-Posedness for Higher-Order Schrödinger Equations in Weighted L2 Spaces

We obtain the global well-posedness for Schrödinger equations of higher orders in weighted L² spaces. This is based on weighted L² Strichartz estimates for the corresponding propagator with higher-order dispersion. Our method is also applied to the Airy equation which is the linear component of Korteweg-de Vries type equations.     

Methods of geometry of differential equations in analysis of integrable models of field theory

In this paper, we investigate algebraic and geometric properties of hyperbolic Toda equations u _xy = exp(Ku) associated with nondegenerate symmetrizable matrices K. A hierarchy of analogues of the potential modified Korteweg-de Vries equation u _t = u _xxx + u ₃ ^x is constructed and its relationship with the hierarchy for the Korteweg-de Vries equation T _t = T _xxx + TT _x is established. Group-theoretic structures for the dispersionless (2 + 1)-dimensional Toda equation u _xy = exp(?u _zz ) are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems Ψ_t = iΨ_xx + i f(|Ψ|) Ψ (multi-soliton complexes) are described.     

Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solutions

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg-de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the x/t plane splits into g+1 soliton regions which are interlaced by g + 1 oscillatory regions, where g + 1 is the number of spectral gaps.     

Shock waves in the modified Korteweg-de Vries-Burgers equation

Summary The asymptotics ast + of shock waves of the modified Korteweg-de Vries-Burgers (MKdV-B) equation is investigated. An attractor interpretation of shock problems for integrable systems is presented and some problems of nonlinear stability are discussed. The MKdV-B equation is considered as a nonconservative perturbation of the integrable modified Korteweg-de Vries (MKdV) equation. The MKdV equation considered here has anon-self-adjoint Lax pair. In spite of this difficulty, acomplex Whitham deformation is constructed.