Existence of Solitary Wave Solutions for the (2 + 1) Dimensional Gardner Equation

在本文中，我们研究了扰动的(2 + 1)维Gardner方程孤立波解的存在性。首先，我们利用平面动力系统的相关知识对未扰系统的平衡点进行研究，得到了孤立波解存在的参数条件并画出了相图。然后，利用几何奇异摄动理论、不变流形理论和弗雷德霍姆理论，通过构造相应微分方程的不变流形，我们得到了相应的同宿轨道。进一步，根据同宿轨道和孤立波之间的对应关系，我们得到了扰动系统孤立波解的存在性。In this paper, we investigate the existence of solitary wave solutions for perturbed (2 + 1) dimensional Gardner equation. Firstly, we utilize the relevant knowledge of planar dynamical systems to study the equilibrium points of undisturbed system, obtain the parameter conditions for the existence of solitary wave solutions, and drew a phase diagram. Then, using geometric singular perturbation theory, invariant manifold theory, and Fredholm theory, we obtain a homoclinic orbit by constructing the invariant manifold of the corresponding differential equation. Further, through the correspondence between homoclinic orbits and solitary waves, we prove the existence of solitary wave solutions for the perturbed system.