Higher Order Korteweg-de Vries Research Articles

The reductive perturbation method and the Korteweg-de Vries hierarchy

By using the reductive perturbation method of Taniuti with the introduction of an infinite sequence of slow time variablesτ1,τ3,τ5, ..., we study the propagation of long surface-waves in a shallow inviscid fluid. The Korteweg-de Vries (KdV) equation appears as the lowest order amplitude equation in slow variables. In this context, we show that, if the lowest order wave amplitudeζ0 satisfies the KdV equation in the timeτ3, it must satisfy the (2n+1)th order equation of the KdV hierarchy in the timeτn+1, withn=2,3,4, ... As a consequence of this fact, we show with an explicit example that the secularities of the evolution equations for the higher-order terms (ζ1,ζ2, ...) of the amplitude can be eliminated whenζ0 is a solitonic solution to the KdV equation. By reversing this argument, we can say that the requirement of a secular-free perturbation theory implies that the amplitudeζ0 satisfies the (2n+ 1)th order equation of the KdV hierarchy in the timeτ2n+1. This essentially means that the equations of the KdV hierarchy do play a role in perturbation theory. Thereafter, by considering a solitary-wave solution, we show, again with an explicit, example that the elimination of secularities through the use of the higher order KdV hierarchy equations corresponds, in the laboratory coordinates, to a renormalization of the solitary-wave velocity. Then, we conclude that this procedure of eliminating secularities is closely related to the renormalization technique developed by Kodama and Taniuti.

Soliton Chain Solutions to the Two-Dimensional Higher Order Korteweg de Vries Equations of Lax Hierarchy

The two-dimensional higher order Korteweg de Vries equations of Lax hierarchy are constructed. Their soliton chain solutions are derived by Hirota's bilinear transformation method, and the various variants of the soliton chain solutions are investigated in some details.

Non-isospectral variable-coefficient higher-order Korteweg-de Vries equations

To the family of nonisospectral variable-coefficient higher-order Korteweg-de Vries equations a new family of nonisospectral variable-coefficient higher-order modified Korteweg-de Vries equations depending on a parametric function eta (t) is constructed. They are connected by an eta 2-dependent Miura transformation. A Backlund transformation is also established. Furthermore, the gauge transformation previously proposed by the authors is applied to this Backlund transformation. For a fixed eta (t), this enables one to derive an autoBacklund transformation for the families of non-isospectral variable-coefficient higher-order eta 2-dependent modified Korteweg-de Vries equations and non-isospectral variable-coefficient higher-order Korteweg-de Vries equations, respectively. As an illustration, three generations of explicit solutions of the non-isospectral second-order variable-coefficient Korteweg-de Vries equation are presented.

Painleve property and pseudopotentials for nonlinear evolution equations

The link between the singularity manifold equation and the pseudopotential equations is found for nonlinear evolution equations both in 1+1 and 2+1 dimensions. Many examples are given: Burgers, Korteweg-de Vries, modified Korteweg-de Vries, Harry Dym, a higher-order Korteweg-de Vries, Sawada-Kotera, Kaup-Kuperschmidt, Kadomtsev-Petviashvili, Boussinesq, modified Kadomtsev-Petviashvili, and (2+1)-dimensional Sawada-Kotera equations.

Gauge and Bäcklund transformations for the variable coefficient higher-order modified Korteweg–de Vries equation

A family of higher-order modified Korteweg–de Vries equations with variable coefficients (t-ho-mKdV) is introduced. A one-to-one correspondence between a real solution of these equations and a complex solution of the variable coefficient higher-order Korteweg–de Vries (t-ho-KdV) equations is established through a complex Miura transformation. An auto-Bäcklund transformation for these t-ho-mKdV equations is derived from that of the t-ho-KdV equations. The associated gauge transformations of the corresponding AKNS systems are presented. They enable one to construct a hierarchy of solutions of the t-ho-mKdV equations from a known hierarchy without solving the differential equations for the wave functions except the first one. A new family of higher-order evolution equations with an auto-Bäcklund transformation is also derived in connection with the gauge transformation of the t-ho-mKdV equations.

The generalized Korteweg-de Vries equation for interacting phonons in a coherent state basis

The higher order Korteweg-de Vries (KdV) equation of interacting phonons is obtained for a one-dimensional anharmonic lattice where fifth order anharmonicity in the potential is taken and justified from the classical continuum approximation method. The possible solitary wave solutions are discussed.

Lattice equations, hierarchies and Hamiltonian structures

By applying a suitable continuum limit to a lattice version of the Korteweg-de Vries (KdV) equation we obtain an infinite hierarchy of integrable partial difference equations for time-dependent fields, defined on the sites of a one-dimensional chain, together with the associated Hamiltonian structure. A second continuum limit applied to any member of the hierarchy yields the well-known hierarchy of higher-order KdV equations with the recursion operator, as well as the Hamiltonian structure. A similar procedure is worked out for the hierarchies related to the modified Korteweg-de Vries (mKdV) equation, starting from a lattice version of the mKdV. For all the equations a direct linearization is given of an integral equation with arbitrary measure and contour.

Bilinearization of Nonlinear Evolution Equations. III. Bäcklund Transformations of Higher-Order Korteweg-de Vries Equations

The Backlund transformations of the higher-order Korteweg-de Vries (KdV) equations are derived on the basis of the bilinear transformation method. It is shown that these transformations are closely related to the quantities obtained from the functional derivative of conserved quantities of the modified KdV equation. The inverse scattering transforms of the higher-order KdV equations are also presented.

Bilinearization of Nonlinear Evolution Equations

A new method for transforming the Lax hierarchy of nonlinear evolution equations u t =-(∂/∂ x )(δ I n /δ u ) into bilinear forms is developed on the basis of the bilinear transformation method, and the structure of these equations is clarified, where I n is the n -th conserved quantity of some nonlinear evolution equation and δ/δ u denotes the functional derivative. The explicit calculations are carried out in the case of the higher-order Korteweg-de Vries equations.

The Bäcklund transformations of the higher-order Korteweg-de Vries equations

The Bäcklund transformations, the conserved quantities and the inverse scattering transforms of the higher-order Korteweg-de Vries equations are presented. It is shown that these Bäcklubd transformations are closely related to the quantities derived from the functional derivative of the conserved quantities of the modified Korteweg-de Vries equation.

N-Soliton Solution of the Higher Order Wave Equation for a Fluid of Finite Depth

Exact N -soliton solution of the higher order wave equation for a stratified fluid layer of finite depth is obtained by employing Hirota's bilinear transformation method. The solution reduces to the N -soliton solution of the higher order Korteweg-de Vries equation in the shallow water limit and to that of the higher order Benjamin-One equation in the deep water limit. The asymptotic form of the solution for large time is derived.

Relations among generalized Korteweg–deVries systems

This report presents certain relations among the completely integrable Hamiltonian systems introduced by Gel’fand and Dikii. These relations generalize a formula of A. Lenard linking the higher-order Korteweg–deVries equations, of which the Gel’fand–Dikii systems are a generalization. The general form of the relations, which connect the various isospectral deformations of linear differential operators, is described, and two examples are given explicitly.

The prolongation structure of a higher order Korteweg-de Vries equation

We have used Wahlquist & Estabrook’s prolongation method to construct a Lie algebra for the higher order Korteweg de Vries (K.deV.) equation q 4 x + α qq 2 x + β q 2 x + γ q 3 ) x + q t = 0 with ( q nx =