Infinitely Many Solitary Waves Due to the Second-Harmonic Generation in Quadratic Media

In this paper, we consider the following coupled Schrödinger system with χ(2) nonlinearities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\{ \begin{array}{l} - \Delta {u_1} + {V_1}\left( x \right){u_1} = \alpha {u_1}{u_2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in {^N}, \\ - \Delta {u_2} + {V_2}\left( x \right){u_2} = \frac{\alpha }{2}u_1^2 + \beta u_2^2,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in {^N}, \\ \end{array} \right.$\end{document} which arises from second-harmonic generation in quadratic media. Here V1(x) and V2(x) are radially positive functions, 2 ≤ N < 6, α > 0 and α > β. Assume that the potential functions V1(x) and V2(x) satisfy some algebraic decay at infinity. Applying the finite dimensional reduction method, we construct an unbounded sequence of non-radial vector solutions of synchronized type.