Doubly periodic solutions of the coupled scalar field equations (Q1772636)

Seventeen kinds of doubly periodic solutions of the coupled Schrödinger-Boussinesq equations \[ \frac{d^2\psi}{d^2\xi}= \alpha_1\psi+\alpha_2\psi\phi,\qquad \frac{d^2\phi}{d^2\xi}= \beta_1\phi+\beta_2\phi^2+\beta_3\psi^2, \] are presented in closed form. The author introduces an auxiliary ordinary differential equation and uses its Jacobi elliptic function solution to construct doubly periodic solutions of the coupled equations. When the module \(m\to 1\), then these solutions degenerate to exact solitary wave solutions of the coupled equations.