An extension of the Super KdV integrable system in terms of operator valued functions is obtained. Following the ideas of Gardner, a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented. The approach is applied to the above described system and infinitely many conserved quantities are constructed. In a particular case they reduce to ...

Algebro-geometric Poisson brackets for real, finite-zone solutions of the Korteweg–de Vries (KdV) equation were studied in [1]. The transfer of this theory to the Toda lattice and the sinh-Gordon equation is more or less obvious. The complex part of the finite-zone theory for the nonlinear Schrödinger equation (NS) and the sine-Gordon equation (SG) is analogous to KdV, but conditions that solut...

We prove in this paper the stability and asymptotic stability in H of a decoupled sum of N solitons for the subcritical generalized KdV equations ut + (uxx + u )x = 0 (1 < p < 5). The proof of the stability result is based on energy arguments and monotonicity of local L norm. Note that the result is new even for p = 2 (the KdV equation). The asymptotic stability result then follows directly fro...

We prove that one system of coupled KdV equations, recently claimed by Hirota, Hu and Tang [J. Math. Anal. Appl. 288:326–348 (2003)] to pass the Painlevé test for integrability, actually fails the test at the highest resonance of the generic branch. Introduction In Section 6 of their recent work [1], Hirota, Hu and Tang reported that the system of coupled KdV equations ∂ui ∂t + 6a ( N

Several types of Darboux transformations for supersymmetric integrable systems such as the Manin-Radul KdV, Mathieu KdV and SUSY sine-Gordon equations are considered. We also present solutions such as supersolitons and superkinks. ⋆ On leave of absence from Beijing Graduate School, CUMT, Beijing 100083, China ⋆⋆ Supported by Beca para estancias temporales de doctores y tecnólogos extranjeros en...

A typical and effective way to construct a higher dimensional integrable equation is to extend the Lax pair for a (1 + 1) dimensional equation known as integrable to higher dimensions. Here we construct an alternative modified KdV equation in (2+1) dimensions by the higherdimensional extension of a Lax pair. And it is shown that this higher dimensional modified KdV equation passes the Painlevé ...

The KdV equation is the canonical example of an integrable nonlinear partial differential equation supporting multi-soliton solutions. Seeking to understand the nature of this interaction, we investigate different ways to write the KdV 2-soliton solution as a sum of two or more functions. The paper reviews previous work of this nature and introduces new decompositions with unique features, putt...

The Hamiltonian-Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem JLu = λu, where J is skewsymmetric and L is self-adjoint. If J has a bounded inverse the index is well-established, and it is given by the number of negative eigenvalues of the operator L constrained to act on some finite-codimensional ...

In this paper, a new space-time spectral algorithm is constructed to solve the generalized Hirota-Satsuma coupled Korteweg-de Vries (GHS-C-KdV) system of time-fractional order. The present algorithm consists of applying the collocationspectral method in conjunction with the operational matrix of fractional derivative for the double Jacobi polynomials, which will be employed as a basis function ...

Long's equation describes stationary flows to all orders of nonlinearity and dispersion. Dissipation is neglected. In this paper, Long's equation is used to attempt to model the propagation of a solibore -a train of internal waves in shallow water at the deepening phase of the internal tide. 1. The Solibore Phenomenon The internal tide in shallow water often has a sawtooth shape rather than a s...