In this paper, we derive a Bäcklund transformation for the supersymmetric Kortwegde Vries equation. We also construct a nonlinear superposition formula, which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea, Ramani and Grammaticos. The celebrated Kortweg-de Vries (KdV) equation was extended into super framework by Kupershmidt [3] in 1984....

In this paper, we derive a Bäcklund transformation for the supersymmetric Korteweg-de Vries equation. We also construct a nonlinear superposition formula, which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea, Ramani and Grammaticos. The celebrated Korteweg-de Vries (KdV) equation was extended into super framework by Kupershmidt [3] in 19...

In this paper we present an overview of the direct relation between differential invariants of projective type for curves in flat semisimple homegenous spaces and PDEs of KdV type. We describe the progress in the proof of a conjectured Theorem stating that for any such space there are geometric evolutions of curves that induce completely integrable evolutions on their invariants of projective-t...

We derive the modulation equations or Whitham equations for the Camassa– Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir...

The N = 2 supersymmetric KdV equations are studied within the framework of Hirota’s bilinear method. For two such equations, namely N = 2, a = 4 and N = 2, a = 1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bäcklund transformation is given for the N = 2, a = 1 supersymmetr...

We consider in this paper the well-posedness for the Cauchy problem associated to two-dimensional dispersive systems of Boussinesq type which model weakly nonlinear long wave surface waves. We emphasize the case of the strongly dispersive ones with focus on the “KdV-KdV” system which possesses the strongest dispersive properties and which is a vector two-dimensional extension of the classical K...

We present a fundamental solution to an initial value problem for the KdV-Whitham system in an explicit integral form. Monotonically decreasing initial data with finite number of breaking points are considered. Generating function for the commuting flows of the averaged KdV hieararchy producing the analytical solutions to the KdV-Whitham system is constructed.

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference betwe...

Head-on collision and overtaking collision between a KdV solitary wave and an envelope solitary wave are first studied in present paper by using Particle-in-cell (PIC) method in a dusty plasma. There are phase shifts of the KdV solitary wave in both head-on collision and the overtaking collision, while no phase shift is found for the envelop solitary wave in any cases. The remarkable difference...

The Korteweg-de Vries (KdV) equation is first derived from a general system of partial differential equations. An analysis of the linearized KdV equation satisfied by the higher order amplitudes shows that the secular-producing terms in this equation are the derivatives of the conserved densities of KdV. Using the multi-time formalism, we prove that the propagation on very long distances is gov...