Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value problem associated with a coupled system consisting of modified Korteweg-de Vries equations for given data. Furthermore, we prove that the unique solution belongs to Gevrey space $G^{\sigma }\times G^{\sigma }$ in x and $G^{3\sigma }\times G^{3\sigma }$ in t. This article is a continuation of recent studies reflected.

The task of our work is to consider the initial value problem based on the model of the generalized Kadomtsev-Petviashvili I equation and prove the local well-posedness in an anisotropic Gevrey spaces and then global well-posedness which improves the recent results on the well-posedness of this model in anisotropic Sobolev spaces ^[17]. Also, wide information about the regularity of the solution in the time variable is provided.

In this article we investigate the coupled system of gKdV equations. A new topological approach is applied to prove the existence of at least one classical