New explicit solutions of Liouville and sine-Gordon equations (Q2703349)

The author constructs explicit solutions to the Liouville equation \(\square u +\lambda\exp u=0\) and the sine-Gordon equation \(\square u+\sin u=0\). Using the generalized symmetry reduction method the following family of solutions of the Liouville equation is obtained \(u=\ln\{(-{h_1(\omega)\over 2\lambda})\text{ sec}^2[{\sqrt{-h_1(\omega)}\over 2}(x_3+h_2 (\omega))]\}\), \((h_1(\omega)<0\), \(\lambda>0)\); \(u=\ln \{{2h_1(\omega)h_2(\omega)\exp(\sqrt{h_1(\omega)}x_3)\over \lambda[1-h_2(\omega)\exp(\sqrt{h_1(\omega)}x_3)]}\}\), \((h_1(\omega)>0\), \(\lambda h_2(\omega)>0)\); \(u=-\ln(\sqrt{\lambda/2} x_3+h(\omega))^2\), where \(h_1(\omega),h_2(\omega),h(\omega)\) are arbitrary two times differentiable functions, and \(\omega\) is an arbitrary solution of the system NEWLINE\[NEWLINE{\partial^2\omega\over\partial x_0^2}- {\partial^2\omega\over\partial x_{L+1}^2}-\ldots- {\partial^2\omega\over\partial x_{n}^2}=0, \quad \left({\partial\omega\over\partial x_0}\right)^2- \left({\partial\omega\over\partial x_{L+1}}\right)^2-\ldots- \left({\partial\omega\over\partial x_{n}}\right)^2=0.NEWLINE\]