Existence and Instability of Standing Wave for the Two-wave Model with Quadratic Interaction

In this paper, we establish the existence and instability of standing wave for a system of nonlinear Schr\"{o}dinger equations arising in the two-wave model with quadratic interaction in higher space dimensions under mass resonance conditions. Here, we eliminate the limitation for the relationship between complex constants $a_{1}$ and $a_{2}$ given in \cite{HOT}, and consider arbitrary real positive constants $a_{1}$ and $a_{2}$. First of all, according to the conservation identities for mass and energy, using the so-called virial type estimate, we obtain that the solution for the Cauchy problem under consideration blows up in finite time in $H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})$ with space dimension $N\geq 4$. Next, for space dimension $N$ with $4<N<6$, we establish the existence of the ground state solution for the elliptic equations corresponding to the nonlinear Schr\"{o}dinger equations under the frequency and mass resonance by adopting variational method, and further achieve the exponential decay at infinity for the ground state. This implies the existence of standing wave for the nonlinear Schr\"{o}dinger equaitons under consideration. Finally, by defining another constrained minimizing problems for a pair of complex-valued functions, a suitable manifold, referring to the characterization of the standing wave, making appropriate scaling and adopting virial type estimate, we attain the instability of the standing wave for the equations under frequency and mass resonance in space dimension $N$ with $4<N<6$ by virtue of the conservations of mass and energy. Here, we adopt the equivalence of two constrained minimizing problems defined for pairs of complex-valued and real-valued functions $(u,v)$, respectively, when $(u,v)$ is a pair of real-valued functions.