For the L subcritical and critical (gKdV) equations, Martel [11] proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N solitons asymptotically as t → +∞. More recently, for the L supercritical case, Côte, Martel and Merle [4] proved the existence of at least one multi-soliton. I...

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1 + 1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painlevé test, is presented. A transformation that links this equation to the canonical form of the Calogero–Bogoy...

In this paper we consider the slightly L-supercritical gKdV equations ∂tu + (uxx + u|u|)x = 0, with the nonlinearity 5 < p < 5 + ε and 0 < ε ≪ 1 . In the previous paper [10] we know that there exists an stable selfsimilar blow-up dynamics for slightly L-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of ...

In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV ...

In this paper, we show the global well-posedness for periodic gKdV equations in the space H(T), s ≥ 1 2 for quartic case, and s > 5 9 for quintic case. These improve the previous results of Colliander et al in 2004. In particular, the result is sharp in the quartic case.

This work is devoted to the study of a higher order numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p = 4) in a bounded domain. The KdV equation and some of its generalizations as the GKdV type equations appear in Physics, for example in the study of waves on shallow water. Based on the analysis of stability of the first order scheme introduced by Pazoto et a...

In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. order to bypass analytical ill-posedness of equation in Sobolev support measures, establish deterministic well-posedness gauged gKdV within framework Fourier-Lebesgue spaces. Our argument relies on bilinear and trilinear Strichartz estimates adapted setti...

In this paper, we prove that there exist no blow-up solutions of the critical generalized Korteweg–de Vries (gKdV) equation with minimal L2-mass, assuming an L2-decay on the right on the initial data.

In this paper, we discuss the Hamiltonian structure of Korteweg–de Vries equation, modified Korteweg–de Vries equation, and generalized Korteweg– de Vries equation. We proposed the Sine-function algorithm to obtain the exact solution for non-linear partial differential equations. This method is used to obtain the exact solutions for KdV, mKdV and GKdV equations. Also, we have applied the method...

&lt;p style='text-indent:20px;'&gt;We establish a smoothing result for the generalized KdV (gKdV) on torus with polynomial non-linearity, damping, and forcing that matches level gKdV at &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ H^1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. As consequence, we existence of global attractor this equation as well its compactness...