Abstract: The questions about soliton resolution, soliton stability or formation of blow-up in KdV-type equations have been intriguing the researchers for quite some time. In this talk we will look at a higher dimensional version of the KdV equation, called Zakharov-Kuznetsov (ZK) equation and discuss behavior of solutions in the 2d and 3d models as those are physically relevant. In particular, we will examine the behavior of solutions close to solitons in different settings. Direct numerical simulations for the KdV-type equations, such as ZK, with generic data show that solutions split into solitons traveling in the positive x-direction and radiation dispersing in the negative x-direction (possibly at a specific angle in dimension 2 and higher). In the L^2-critical and supercritical cases (for example, 2d cubic ZK equation), some of the solitons, traveling to the right, blow-up in finite time. Analytically, we prove existence of blow-up solutions in the 2d cubic (critical) ZK equation. In subcritical case, such as 3d quadratic ZK, we obtain asymptotic stability of solitons in finite energy space. The talk is based on joint works with Luiz Gustavo Farah, Justin Holmer, Christian Klein, Nikola Stoilov, and Kai Yang.
Please access the talk through the Research Seminars page ( https://researchseminars.org/seminar/MO_Analysis ); you will need to register for a free account. If you have any difficulty, please contact Samuel Walsh (firstname.lastname@example.org).