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 Schrödinger equations arising in the two-wave model with quadratic asymmetric interaction in higher space dimensions under mass resonance conditions. Here, we relax the limitation for the relationship between complex constants a 1 and a 2 given in Hayashi et al. (Ann L’inst Henri Poincaré-AN 30:661–690, 2013), and consider arbitrary real positive constants a 1 and a 2 . First of all, according to the conservation quantities of mass and energy, using the so-called virial type estimate, we obtain that the solution of the Cauchy problem under consideration blows up in finite time in H 1 ( R N ) × H 1 ( R N ) with space dimension N ≥ 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 Schrödinger equations under frequency and mass resonances by adopting variational methods, and further achieve the exponential decay at infinity for the ground state. This implies the existence of standing wave for the nonlinear Schrödinger equaitons under consideration. Finally, by defining another constrained minimizing problem 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 resonances 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.