Article
Ufa Mathematical Journal
Volume 3, Number 3, pp. 65-77
Goursat problem for nonlinear hyperbolic systems with integrals of the first and second order
Kostrigina O.S., Zhiber A.V.
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We consider the Goursat problem for a one class of nonlinear hyperbolic systems of equations of the form $$u^i_{xy}=F^i(u, u_x, u_y),\ i=1,2, \ u=(u^1,u^2)$$ with integrals of the first and second order \begin{gather*} \omega^1(u^1,u^2,u^1_x,u^2_x), \ \omega^2(u^1,u^2,u^1_x,u^2_x,u^1_{xx},u^2_{xx}), \ (\bar{D}(\omega^1)=\bar{D}(\omega^2)=0),\\ \bar{\omega}^1(u^1,u^2,u^1_y,u^2_y), \ \bar{\omega}^2(u^1,u^2,u^1_y,u^2_y,u^1_{yy},u^2_{yy}), \ (D(\bar{\omega}^1)=D(\bar{\omega}^2)=0). \end{gather*} Explicit formulas for the solutions of the Goursat problem with the data set on the characteristics \begin{gather*} u^1(x_0,y)=\phi_1(y), \ \ u^2(x_0,y)=\phi_2(y), \\ u^1(x,y_0)=\psi_1(x), \ \ u^2(x,y_0)=\psi_2(x) \end{gather*} are obtained.