Nonlinear d'Alambert equation in the pseudo-Euclidean space \(\mathbb R_{2,n}\) and its solutions (Q2740352)

The author considers the equation of the form NEWLINE\[NEWLINE\square u+F(u)=0\tag{1}NEWLINE\]NEWLINE where \(\square u=u_{11}+u_{22}-u_{33}- \cdots -u_{n+2,n+2}\), \(u_{ij}:=\partial u/\partial x_i\partial x_j\). Using the method suggested in the paper of \textit{A. F. Barannik} and \textit{I. I. Yurik} [Ukr. Math.J. 51, No. 5, 649-661 (1999); translation from Ukr. Mat. Zh. 51, No. 5, 583-593 (1999; Zbl 0974.35507)], new classes of exact solutions to (1) are constructed. NEWLINENEWLINENEWLINEThe basic idea is as follows. Suppose that a symmetry ansatz for the equation (1) has the form NEWLINE\[NEWLINEu=f(x)\varphi(\omega_1,\ldots,\omega_k)+g(x)\tag{2}NEWLINE\]NEWLINE where \(\omega_j(x_1,\ldots,x_{n+2})\) are new independent variables. Then the new ansatz NEWLINE\[NEWLINEu=f(x)\varphi(\omega_1,\ldots,\omega_k,\omega _{k+1},\ldots,\omega _{l})+g(x)\tag{3}NEWLINE\]NEWLINE is sought under appropriate choice of functions \(\omega _{k+1},\ldots,\omega_l\) in such a way that the reduced equation corresponding to ansatz (3) has the same form as that corresponding to the ansatz (2). Using this idea the author constructs 7 ansatzes and corresponding exact solutions to (1). Special cases when \(F=\lambda \exp u\) and \(\lambda u^k\) are analyzed.