which can be considered as a coupling between the KdV (with respect to u) and the mKdV (with respect to v) equations. The coupled KdV-mKdV equations were proposed by Kersten and Krasil’shchik [1] and originate from a supersymmetric extension of the classical KdV [2]. It also can be considered as a coupling between the KdV and mKdV equations: By setting v = 0 we obtain the KdV equation ut + uxxx...

In this work, we develop two new (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation and negative-order KdV-CBS (nKdV-nCBS) equation. The newly developed equations pass the Painlevé integrability test via examining compatibility conditions for each model. We examine dispersion relation derive multiple soliton solutions

We consider the logarithmic Korteweg–de Vries (log–KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in H(R) with conserved L norm and energy, we construct a weak global solution of the log–KdV equation in a subset of H(R). This construction yields c...

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...

Based on estimates for the KdV equation in analytic Gevrey classes, a spectral collocation approximation of the KdV equation is proved to converge exponentially fast. Mathematics Subject Classification. 35Q53, 65M12, 65M70. Received: March 31, 2006. Revised: July 11, 2006.