multidimensional complex nonlinear d'Alembert equation
On some class of non-Lie solutions of nonlinear d'Alambert equation
The authors construct a new class of the exact solutions of the multidimensional complex nonlinear d'Alambert equation \((\partial^{2}/\partial x_{0}^{2} -\Delta)u=F(|u|)u\), where \(F\) is some continuous function; \(u=u(x_{0},x_{1},x_{2},x_{3})\). This class can not be obtained within the framework of the traditional Lie approach. For the equation \((\partial^{2}/\partial x_{0}^{2}-\Delta)u=\lambda|u|^{2}u\) this class of solutions has the form \(u(x)=(1/\lambda)w_{1}(\omega)\exp\{iw_{1}(\omega)x_{3}+ iw_{2}(\omega)\}\), \(\omega=(x_{1}^{2}+x_{2}^{2})^{-1} (x_{0}x_{1}\pm x_{2}\sqrt{x_{1}^{2}+x_{2}^{2}-x_{0}^{2}})\), where \(w_{1},w_{2}\) are arbitrary functions.
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