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...

In this paper, we establish exact solutions for the timefractional Klein-Gordon equation, and the time-fractional HirotaSatsuma coupled KdV system. The Hes semi-inverse and the Kudryashov methods are used to construct exact solutions of these equations. We apply Hes semi-inverse method to establish a variational theory for the time-fractional Klein-Gordon equation, and the timefractional Hirota...

In this Letter, we analyze the structure of linearization operators of the Korteweg–de Vries (KdV) hierarchy equations expanded around single-soliton solutions. We uncover the remarkable property that these linearization operators can be factored into the integro-differential operator which generates this hierarchy and the linearization operator of the KdV equation. An important consequence of ...

The matrix 2x2 spectral differential equation of the second order is considered on x in (−∞, +∞). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second covariant equation to form Lax pair of a coupled KdV-MKdV system. The sequence of the elementary Darboux transformations of the zero-potential seed produce two-parameter soluti...

Abstract: In this paper, we studied the numerical solution of a time-fractional Korteweg–de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi–Gauss–Lobatto (JGL) nodes was applie...