Infinitely many elliptic solutions to a simple equation and applications (Q2015770)

The aim of the paper is to show that certain nonlinear PDE could have infinitely many solutions presented in terms of Weierstrass and Jacobi elliptic functions. Main ingredient of the method is a nonlinear iterative formula of solutions. The authors first defined such formula for simple auxiliary differential equation \[ \left(\frac{d\eta(\xi)}{d \xi}\right)^2 = a \eta + b \eta^2 + c \eta^3. \tag{1} \] The nonlinear iterative formula for equation (1) is obtained by using Bäcklund transformation and a modified truncation approach. By using this formula the authors obtained infinitely many Weierstrass and Jacobi solutions. The proposed method is applied to obtain infinitely many Weierstrass and Jacobi solutions to a number of special PDEs, namely \[ (u_t - 6 u u_{xx} + u_{xxx})_x + 3 u_{yy} = 0, \] \[ \left(\frac{u_{xx}}{u}\right)_t + 2 u u_x = 0, \] \[ \begin{aligned} i u_t + u_{xx} - u_{yy} - 2 |u|^2 u - 2 u v &= 0, \\ v_{xx} + v_{yy} + 2 \left(|u|^2\right)_{xx} &= 0. \end{aligned} \]