  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.
