Geodesic
On a class of hyperbolic equations with third-order integrals
Ufa mathematical journal, Tome 15 (2023) no. 2, pp. 20-30

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We consider a Goursat problem on classification nonlinear second order hyperbolic equations integrable by the Darboux method. In the work we study a class of hyperbolic equations with second order $y$-integral reduced by an differential substitution to equations with first order $y$-integral. It should be noted that Laine equations are in the considered class of equations. In the work we provide a second order $y$-integral for the second Laine equation and we find a differential substitution relating this equation with one of the Moutard equations. We consider a class of nonlinear hyperbolic equations possessing first order $y$-integrals and third order $x$-integrals. We obtain three conditions under which the equations in this class possess first order and third order integrals. We find the form of such equations and obtain the formulas for $x$- and $y$-integrals. In the paper we also provide differential substitutions relating Laine equations.
Keywords: $x$- and $y$-integrals, differential substitutions.
Mots-clés : Laplace invariants 
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Yu. G. Voronova; A. V. Zhiber. On a class of hyperbolic equations with third-order integrals. Ufa mathematical journal, Tome 15 (2023) no. 2, pp. 20-30. http://geodesic.mathdoc.fr/item/UFA_2023_15_2_a2/