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.

In order to study the longtime behavior of a dissipative evolutionary equation, we generally aim to show that the dynamics of the equation is finite dimensional for long time. In fact, one possible way to express this fact is to prove that dynamical systems describing the evolutional equation comprise the existence of the global attractor 1 . The KDV equation without dissipative and forcing was...

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually performed on the time scale, for which dynamics of the KdV equation is defined. We show how to extend justification analysis on longer time intervals provided dyn...

We prove that the Cauchy problem of the Schrödinger Korteweg deVries (NLS-KdV) system on T is globally well-posed for initial data (u0, v0) below the energy space H × H. More precisely, we show that the non-resonant NLS-KdV is globally wellposed for initial data (u0, v0) ∈ H (T)× H(T) with s > 11/13 and the resonant NLS-KdV is globally well-posed for initial data (u0, v0) ∈ H (T)×H(T) with s > ...

We study here the water waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known that, for such regimes, a generalization of the KdV equation (somehow linked to the CamassaHolm equation) can be derived and justified [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when...

In this paper the stability of the Korteweg-de Vries (KdV) equation is investigated. It is shown analytically and numerically that small perturbations of solutions of the KdV-equation introduce effects of dispersion, hence the perturbation propagates with a different velocity then the unperturbed solution. This effect is investigated analytically by formulating a differential equation for pertu...