In the present paper, we make a rigorous study of the solitary wave solutions to a coupled Schrödinger system with quadratic and cubic nonlinearity. This kind of system of Schrödinger equations arises from optics theory. First, the existence and nonexistence of nontrivial solutions, respectively, in focusing and defocusing cases are considered. Second, we prove the existence of multiple nontrivial solutions by using the Crandall–Rabinowitz local bifurcation theorems and calculate the exact Morse index of these solutions. Third, the continuous dependence on the parameter and asymptotic behavior of positive ground state solutions in the focusing case are also established. Particularly, from the mathematical point of view, we prove the behavior of positive solution coincides with the physical phenomena of Bang et al. (Opt Lett 22(22):1680–1682, 1997; Phys Rev E (3) 58(4):5057–5069, 1998). Finally, we prove the existence of sign-changing solutions.

中文翻译：

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混合非线性增长的椭圆耦合方程组正解和正则解的存在性。

在本文中，我们对具有二次和三次非线性耦合的Schrödinger系统的孤波解进行了严格的研究。这种薛定ding方程组是由光学理论产生的。首先，分别考虑在聚焦和散焦情况下非平凡解的存在和不存在。其次，我们使用Crandall–Rabinowitz局部分歧定理证明了多个非平凡解的存在，并计算了这些解的精确莫尔斯系数。第三，还建立了在聚焦情况下对正基态解的参数和渐进行为的连续依赖。特别是，从数学角度来看，我们证明了正解的行为与Bang等人的物理现象相吻合。（Opt Lett 22（22）：1680–1682，1997；E（3）58（4）：5057-5069，1998）。最后，我们证明了符号转换解决方案的存在。