Dynamical symmetries and conservation laws for Korteweg-de Vries equation

The KdV-equation in two space time dimensions with the set of rapidly decreasing test functions as initial conditions is treated in the setting of nonlinear group and Lie algebra representations. The topological properties of the direct and inverse scattering mappings are discussed in detail.The algebra of continuous constants of motion turns out to be generated as in the linear case by three constants of motion and an extension of a representation of the e₂ Lie algebra on space-time symmetries to its enveloping algebra. The integrability of these representations is studied.It is further proved that the “moment problem” does not have a unique solution in this setting.The existence of noncommutative algebras of smooth time independent constants of motion is pointed out.     

Lax pair,infinitely-many conservation laws and soliton solutions for a set of the time-dependent Whitham-Broer-Kaup equations for the shallow water

Under investigation in this paper is a set of the time-dependent Whitham–Broer–Kaup equations, which is used for the shallow water under the Boussinesq approximation. The equations can be transformed into generalized time-dependent coefficient Ablowitz–Kaup–Newell–Segur system via the variable transformation. Lax pair, infinitely-many conservation laws and bilinear forms of the Ablowitz–Kaup–Newell–Segur system are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. The solitons are physically related to the horizontal velocity field and height that deviates from equilibrium position of the water. Features of the solitons are studied: Soliton amplitude is related to the wave number parameters, while the soliton velocity is related to the wave number parameters and variable coefficient. Interactions between/among the solitons could be elastic or inelastic, determined by the wave number parameters. Interaction property could not be affected by the variable coefficient. Soliton stability is studied via the numerical calculation, which indicates that the solitons could only propagate steadily in a limited time.     

Infinitely Many Lax Pairs of the Korteweg-de Vries Equation

Starting from a known Lax pair, one can get some infinitely many coupled Lax pairs.In this letter, we take the well-known KdV equation as a typical example. Using infinitely many symmetries, the infinitely many inhomogeneous linear Lax pairs of KdV equation can be obtained. And considering the Darboux transformations for the KdV equation leads to the infinitely many inhomogeneous nonlinear Lax pairs.     

Conservation laws for the Korteweg-de Vries equation and the theory of partitions

We show how the terms appearing in the expressions for the densities and the fluxes for the Korteweg-de Vries equation may be found by combinatorial methods. Our basic device consists in associating partitions and their Ferrers graphs to the first density and to the first flux, and then in proceeding inductively following very simple rules. Furthermore, we use unrestricted partitions and a recurrence relation to specify every term of every integral power of the Sturm-Liouville (or one-dimensional Schrödinger) operator.     

The matrix Lax representation of the generalized Riemann equations and its conservation laws

It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter-Saxton equation. New matrix and scalar Lax representation are presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.     

Group-theoretical interpretation of the Korteweg-de Vries type equations

The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schrödinger equation (with nonlocal potential) plays the same role as the one-dimensional Schrödinger equation does in the theory of the Korteweg-de Vries equation.     

Consistent Riccati expansion solvable classification and soliton-cnoidal wave interaction solutions for an extended Korteweg-de Vries equation

In this paper, we consider an extended Korteweg-de Vries (KdV) equation. Using the consistent Riccati expansion (CRE) method of Lou, the extended KdV equation is proved to be CRE solvable in only two distinct cases. These two CRE solvable models are the KdV-Lax and KdV-Sawada-Kotera (KdV-SK) equations. In addition, applying the nonauto-Bäcklund transformations which are provided by the CRE method, we present the explicit construction for soliton-cnoidal wave interaction solutions which represent a soliton propagating on a cnoidal periodic wave background in the KdV-Lax and KdV-SK equations, respectively.     

Generation of large-amplitude solitons in the extended Korteweg-de Vries equation

We study the extended Korteweg-de Vries equation, that is, the usual Korteweg-de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg-de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude "table-top" solitons as well as small-amplitude solitons similar to the solitons of the Korteweg-de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a "sech"-profile. (c) 2002 American Institute of Physics.     

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

In this paper, we put our focus on a variable-coe~cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.     

Korteweg-de Vries hierarchy using the method of base equations

A base-equation method is implemented to realize the hereditary algebra of the Korteweg-de Vries (KdV) hierarchy and the N-soliton manifold is reconstructed. The novelty of our approach is that, it can in a rather natural way, predict other nonlinear evolution equations which admit local conservation laws. Significantly enough, base functions themselves are found to provide a basis to regard the KdV-like equations as higher order degenerate bi-Lagrangian systems.     

Rational solutions of the Korteweg-de Vries equation in wronskian form

The rational solutions of the Korteweg-de Vries equation are obtained as limits of the soliton solutions in wronskian form and these solutions are verified by direct substitution using a novel determinantal identity.     

The complexiton solutions to the nonisospectral Korteweg-de Vries equation

Starting with the solution classification for a linear differential equations, the complexiton solutions to nonisospectral Korteweg-de Vries equation are presented. The basic technique adopted is the Wronskian technique for Hirota's bilinear equation.     

Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations

In this paper we establish new results about the existence, stability, and instability of periodic travelling wave solutions related to the critical Korteweg-de Vries equation

u_t+5u⁴u_x+u_xxx=0,     

Integrability and solutions of a nonsymmetric discrete Korteweg-de Vries equation

In this paper, we present Lax pairs and solutions for a nonsymmetric lattice equation, which is a torqued version of the lattice potential Korteweg-de Vries equation. This nonsymmetric equation is special in the sense that it contains only one spacing parameter but consists of two consistent cubes with other integrable lattice equations. Using such a multidimensionally consistent property we are able to derive its two Lax pairs and also construct solutions using B?cklund transformations.     

The integration of Burgers and Korteweg-de Vries equations with nonuniformities

In this letter we demonstrate that both Burgers and Korteweg-de Vries equations with nonuniformity terms can be reduced to a Burgers or Korteweg-de Vries equation with constant coefficients if these terms satisfy a compatibility condition.     

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found. Received 15 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: nakostov@ie.bas.bg     

Exact solutions of the Korteweg-de Vries equation with a self-consistent source

Exact solutions are found for the Korteweg-de Vries equation with a source satisfying the stationary Schrödinger equation. Each solution describes the evolution of the initial moving wave with one phase velocity to the final moving wave with another phase velocity. The conditions are pointed out under which the phase velocities of these waves may differ in sign. The obtained results are relevant to some problems of hydrodynamics, plasma physics, solid state physics, etc.