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.
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