Collocation method using quintic B-splines finite element have been developed for solving numerically the HirotaSatsuma coupled MKdV equation. Accuracy of the proposed method is shown numerically by calculating conservation laws, 2 L and  L norms on studying of a soliton solution. It is shown that the collocation scheme for solutions of the MKdV equation gives rise to smaller errors and is qui...

In this paper we find a explicit moving frame along curves of Lagrangian planes invariant under the action of the symplectic group. We use the moving frame to find a family of independent and generating differential invariants. We then construct geometric Hamiltonian structures in the space of differential invariants and prove that, if we restrict them to a certain Poisson submanifold, they bec...

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for H initial data, s > −1/2, and for any s1 < min(3s+1, s+ 1), the difference of the nonlinear and linear evolutions is in H1 for all times, with at most polynomially growing H1 norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case s ≥ 0. Our resu...

We found a new symplectic structure and a recursion operator for the Sasa–Satsuma (complex mKdV II) equation widely used in nonlinear optics, pt = pxxx + 6pqpx + 3p(pq)x, qt = qxxx + 6pqqx + 3q(pq)x, along with an integro-differential substitution linking this system to a third-order generalized symmetry of the complex sine-Gordon II system uxy = vuxuy uv + c + (2uv + c)(uv + c)ku, vxy = uvxvy ...

in this paper, we have studied on the solutions of modied kdv equation andalso on the stability of them. we use the tanh method for this investigationand given solutions are good-behavior. the solution is shock wave and can beused in the physical investigations

1. As was shown in the remarkable communication [4] the Cauchy problem for the Korteweg–de Vries (KdV) equation ut = 6uux−uxxx, familiar in theory of nonlinear waves, is closely linked with a study of the spectral properties of the Sturm–Liouville operator Lψ = Eψ, where L = −d/dx + u(x). For rapidly decreasing initial conditions u(x, 0), where ∫∞ −∞ u(x, 0)(1 + |x|)dx < ∞, the KdV equation can...

The higher order wave equation of KdV type, which describes many important physical phenomena, has been investigated widely in last several decades. In this work, multisymplectic formulations for the higher order wave equation of KdV type are presented, and the local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-symplectic discretization of e...

A coupled KdV system with a free parameter proposed by Nutku and Og̃uz is considered. It is shown that the system passes the WTC’s Painlevé test for arbitrary value of the parameter. A further analysis yields that the parameter can be scaled away and the system can be decoupled. Soliton or integrable equations are those systems which have rich structures, such as infinite number of conservation ...

The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi’s elliptic function expansion. presented have a more general form than those studied in literature. Nine couples found. Each couple is symmetric mathematical form. In limit cases m→1, these periodic degenerate as corresponding soliton solutions. After simple parame...