Abstract. We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlevé I hierarchy. It is shown that this solution gives a valid asympto...

Recent studies of the evolution of weakly nonlinear long waves in shear flows have revealed that when the wave field contains a critical layer, a new nonlinear wave equation is needed to describe the wave evolution. This equation is of the same type as the well-known Korteweg-de Vries equation but has a more complicated nonlinear structure. Our main interest is in the steady solitary wave solut...

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...

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 generation of instability in inviscid, non-di usive geophysical ows is generically caused by a resonance between two wave modes. The weakly nonlinear unfolding of this situation is described in the long-wave regime, using a particular two-layer quasi-geostrophic model as an illustrative example. The outcome is a system of two coupled Korteweg-de Vries equations. This system contains a very ...

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...

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 ...

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 ...

Sufficient conditions are given for stability of solitary-wave solutions of model equations for one-dimensional long nonlinear waves. These conditions differ from others which have appeared previously in that they are phrased in terms of positivity properties of the Fourier transforms of the solitary waves. Their use leads to simplified proofs of existing stability results for the Korteweg-de V...