We consider the critical gKdV equation with a saturated perturbation: , where and . For any initial data , the corresponding solution is always global and bounded in . This equation has a family of solutions, and our goal is to classify the dynamics near solitons. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave whose norm is of size as ; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at ; (iii) the solution leaves any small neighborhood of the modulated family of the solitary waves.
This extends the classification of the rigidity dynamics near the ground state for the unperturbed critical gKdV (corresponding to ) by Martel, Merle and Raphaël. However, the blow-down behavior (ii) is completely new, and the dynamics of the saturated equation cannot be viewed as a perturbation of the critical dynamics of the unperturbed equation. This is the first example of classification of the dynamics near the ground state for a saturated equation in this context. The cases of critical NLS and supercritical gKdV, where similar classification results are expected, are completely open.
"On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation." Anal. PDE 12 (1) 43 - 112, 2019. https://doi.org/10.2140/apde.2019.12.43