The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary d...

The basic theory of internal solitary waves is reviewed, with the main emphasis on applications to the many observations of such waves in shallow coastal seas, fjords, lakes and in the atmospheric boundary layer. Commencing with the equations of motion for an inviscid, incompressible density-stratified fluid, we describe asymptotic reductions to model long-wave equations, including the well-kno...

For the nonlinear Schrödinger equation, the Korteweg-de Vries equation, and the modified Korteweg-de Vries equation, periodic exact solutions are constructed from their stationary periodic solutions, by means of the Bäcklund transformation. These periodic solutions were not written down explicitly before to our knowledge. Their asymptotic behavior when t-->-infinity is different from that when ...

In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots. On leave of absence from Bei...

We show that the smooth traveling waves of the Camassa-Holm equation naturally correspond to traveling waves of the Korteweg-de Vries equation.

We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the domain finite. We specially discuss the soliton solutions of the K(m, n) equation and KdV-K(m, n) equation. Furthermore for the mixed models of linear and nonli...

We discuss several PT -symmetric deformations of superderivatives. Based on these various possibilities, we propose new families of complex PT -symmetric deformations of the supersymmetric Korteweg-de Vries equation. Some of these new models are mere fermionic extensions of the former in the sense that they are formulated in terms of superspace valued superfields containing bosonic and fermioni...

We develop the inverse scattering transform method for the Novikov equation ut − utxx + 4uux = 3uuxuxx + uuxxx considered on the line x ∈ (−∞,∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann–Hilbert (RH) problem, which in this case is a 3×3 matrix problem. The structure of this RH problem shares many common features with the case of ...

Consider a one-dimensional Schrödinger operator which is a shortrange perturbation of a quasi-periodic, finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions o...

or equal to Pc > U(O)*‘, where U(O)* ’ IS the propagation speed of shallow-water waves and Ue is determined by Eq. ( 12). The solitary wave is found by solving a forced Korteweg-de Vries equation. The amplitude of the solitary wave is proportional to the upstream velocity and the downstream elevation is inversely proportional to the upstream velocity. A second branch of solutions of the forced ...