Cauchy Problem for for some high order generalization of Korteweg - de Vries equation

In this work we study Cauchy problem for a high-order differential equation $\frac{\partial u(y,x)}{\partial y}+P(\frac{\partial}{\partial x})u(y,x)=\gamma\frac{\partial}{\partial x}(u^2(y,x))+F(y,x)$. We prove that the problem is well-posed both for linear ($\gamma =0$) and nonlinear equations on the class of rapidly decaying Schwartz functions. Furthermore, for the case when the initial condition is given on $L_2(\mathbf{R}^1)$ we prove the existence of the unique solution on the space $L_{\infty}(0,y_0; L_2(\mathbf{R}^1))\bigcap L_2(0,y_0; H^{n-1}(\mathbf{R}^1))\bigcap L_2(0,y_0;H^{n}(-r, r))$, where $r$ is an arbitrary positive number. It is also shown that the solution continuously depends on the initial conditions.