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On functionally invariant solutions of a class of quasilinear second order partial differential equations with two independent variables. (Russian) Zbl 0637.35021

Necessary and sufficient conditions are imposed on the coefficients of a quasilinear second order partial differential equation \[ U_{tt}+2B(p,q)U_{tx}+C(p,q)U_{xx}=0;\quad p=U_ x,\quad q=U_ t,\quad B^ 2-C>0, \] such that there exist two functionally invariant solutions of the above equation \(U=\Phi_ 1(x+a_ 1t)\); \(U=\Phi_ 2(x+a_ 2t)\), where \(\Phi_ i\) are arbitrary functions, \(a_ i\) are fixed constants \((a_ 1-a_ 2\neq 0\), \(i=1,2).\)
In a particular case the well-known Born-Infeld equation can be obtained \[ [1+(U_ x)^ 2]U_{tt}-2U_ xU_ tU_{xt}+[(U_ t)^ 2- 1]U_{xx}=0, \] which has two functionally invariant solutions: \(U=\Phi_ 1(x-t)\); \(U=\Phi_ 2(x+t)\).
Reviewer: O.F.Men’shikh

MSC:

35G20 Nonlinear higher-order PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
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