Orbital stability of bifurcating solutions in the nonlinear Schrödinger system with quadratic interaction

In this paper, we study the existence of bifurcation solutions and the orbital stability of a class of coupled elliptic systems with quadratic nonlinearities. These systems are relevant to Raman amplification in plasma. Using the Crandall-Rabinowitz local bifurcation theorem, we prove the existence of positive bifurcation solutions. Additionally, we calculate the Morse index and establish the orbital stability and instability of these bifurcation solutions. Notably, we extend our study to the bifurcation results at the bifurcation point, addressing an open question posed by M. Colin, T. Colin, and M. Ohta (Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), 2211–2226; SIAM J. Math. Anal. 44 (2012), 206–223). Furthermore, we discover that the bifurcation solution emerges from the semitrivial solution, indicating a stability exchange when the parameter is small in certain regions. This phenomenon is a particularly interesting finding in this context.