Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard...

1 KdV equation and Schrödinger operator 2 1.1 Integrability of Korteweg – de Vries equation . . . . . . . . . . . . . . . . . . 2 1.2 Elements of scattering theory for the Schrödinger operator . . . . . . . . . . . 5 1.3 Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Dressing operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We study solitary wave interactions in the Euler–Poisson equations modeling ion acoustic plasmas and their approximation by KdV n-solitons. Numerical experiments are performed and solutions compared to appropriately scaled KdV n-solitons. While largely correct qualitatively the soliton solutions did not accurately capture the scattering shifts experienced by the solitary waves. We propose corre...

In this paper, we study the propagation of bores over a long distance. We employ experimental data as input for numerical simulations using COULWAVE. The flume is extended numerically to an effective relative length x/h=3000, which allows all far-field solitons emerge from undular bore in simulation data. apply periodic KdV-based nonlinear Fourier transform (KdV-NFT) time series taken at differ...

Numerical simulations of the interaction internal solitary waves (ISWs) opposite polarity are conducted by solving incompressible Euler equations under Boussinesq approximation. A double-pycnocline stratification is used. method to determine when ISWs both polarities exist also presented. The confirm previous work that same soliton-like; however, here it shown a fast ISW with as Korteweg–de Vri...

We prove that the KdV-Burgers is globally well-posed in H−1(T) with a solution-map that is analytic fromH−1(T) to C([0, T ];H−1(T)) whereas it is ill-posed in Hs(T), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(T) to even D′(T) at any fixed t > 0 small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dis...

We study the relation between the centro-affine geometry of starshaped planar curves and the projective geometry of parametrized maps into RP. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwa...

This paper studies the problem of optimal control of the viscous KdV-Burgers’ equation. We develop a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the viscous KdV-Burgers’ equation. While the viscous KdV-Burgers’ equation is transformed into a simpler linear equation, the performance index is transformed to a complicated rational expression. We show t...

We construct a new variety of N = 2 supersymmetric integrable systems by junction of pseudo-differential superspace Lax operators for a = 4, N = 2 KdV and multi-component N = 2 NLS hierarchies. As an important particular case, we obtain Lax operator for N = 4 super KdV system. A similar extension of one of N = 2 super Boussinesq hierarchies is given. We also present a minimal N = 4 supersymmetr...

The KdV equation arises in the framework of the Boussinesq scaling as a model equation for waves at the surface of an inviscid fluid. Encoded in the KdV model are relations that may be used to reconstruct the velocity field in the fluid below a given surface wave. In this paper, velocity fields associated to exact solutions of the KdV equation are found, and particle trajectories are computed n...