Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation...

and Applied Analysis 3 hard to obtain because they are highly nonlinear equations and most probably they are not integrable equations in general. Thus, large numbers of research results are still concentrated in the classical KdV equation and some other high-order equations with KdV type, such as KdV-Burgers equation [17, 18] and KdV-Burgers-Kuramoto equation [19], at present. Therefore, the in...

In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevé waves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explic...

A modified Korteweg-de Vries (KdV) equation is obtained for studying the propagation of nonlinear compressional waves and pulses in a chain of particles including the effect of damping. Suitably altering the linear phase velocity makes this equation useful also for the problem of phonon propagation in a two-dimensional (2D) lattice. Assuming a Yukawa potential, we use this method to model compr...

where m, n are the discrete independent variables, vn is the dependent variable on the lattice site (m, n), and am, bn are arbitrary functions of m and n, respectively. Recently eq.(1) has been derived by Matsuura[1] as the equation of motion of discrete curves on the centro-affine plane. In this context, vn is related to the curvature and it is essential that am and bn depend on m and n, respe...

The Rotation-Modified Korteweg-deVries (RMKdV) equation differs from the ordinary KdV equation only through an extra undifferentiated term due to Coriolis force. This article describes the steadily-travelling, spatially-periodic solutions which have peaks of identical size. These generalize the “cnoidal” waves of the KdV equation. There are five overlapping regimes in the parameter space. We de...

We present the results of further analysis of the integrability properties of the N = 4 supersymmetric KdV equation deduced earlier by two of us (F.D. & E.I., Phys. Lett. B 309 (1993) 312) as a hamiltonian flow on N = 4 SU(2) superconformal algebra in the harmonic N = 4 superspace. To make this equation and the relevant hamiltonian structures more tractable, we reformulate it in the ordinary N ...

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, φxt + φxxxz/4 + φxφxz + φxxφz/2 + ∂ −1 x φzzz/4 = 0. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modifie...

A method of connecting the Korteweg~e Vries (KdV) equation, known from the theory of nonlinear waves, with the SchrOdinger equation was discovered in 1967,Pl This method is applied in the present paper to a study of a periodic problem. We find exact analytical formulae for a class of solutions u (x ,t) such that at any moment in time t the potential u (x ,t) of the Schrodinger operator has only...

The Gardner equation is an extension of the Korteweg–de Vries (KdV) equation. It exhibits basically the same properties as the classical KdV, but extends its range of validity to a wider interval of the parameters of the internal wave motion for a given environment. In this paper, we derive exact solitary wave solutions for the generalized Gardner equation that includes nonlinear terms of any o...