This article is cited in 6 papers

Goursat problem for nonlinear hyperbolic systems with integrals of the first and second order

A. V. Zhiber^a, O. S. Kostrigina<sup>b

a</sup> Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia
^b Ufa State Aviation Technical University, Ufa, Russia

Abstract: We consider the Goursat problem for one class of nonlinear hyperbolic systems of equations of the form
$$ u^i_{xy}=F^i(u, u_x, u_y),\qquad i=1,2,\quad u=(u^1,u^2), $$
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}),\quad(\overline D(\omega^1)=\overline D(\omega^2)=0),\\ \overline\omega^1(u^1,u^2,u^1_y,u^2_y),\ \overline\omega^2(u^1,u^2,u^1_y,u^2_y,u^1_{yy},u^2_{yy}),\quad(D(\overline\omega^1)=D(\overline\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),\quad u^2(x_0,y)=\phi_2(y),\\ u^1(x,y_0)=\psi_1(x),\quad u^2(x,y_0)=\psi_2(x). \end{gather*}


Keywords: nonlinear hyperbolic equations, characteristics, Goursat problem.

UDC: 517.9

Received: 15.07.2011

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