On the nonlinear Dysthe equation

Strichartz and Multi-linear Estimates for the One-dimensional Periodic Dysthe equation

“…Motivations for such ill-posedness and conjectures of stronger results can be found in Section 5. Similar results were shown recently in [8] for the Dysthe equation in different spatial dimensions and settings. 1.5.…”

“…This suggests that any energy methods in H s space should fail for the 1D periodic Dysthe equation. By some more delicate construction, it is expectable to obtain an ill-posedness result similar as in [8].…”

“…Then we compute the contributions to F x u 3 (n) for each nonlinear term in N : 3 2 |u| 2 ∂ x u; i 2 u|∂ x ||u| 2 ; and 1 4 u 2 ∂ x u * . We disregard the nonlinearity |u| 2 u since it does not have a significant contribution as shown in [8].…”

“…The problem differs from that framework of the whole-space: in the whole-space setting, the Dysthe equation exhibits dispersive behaviors (see e.g. [8]), whereas solutions in the periodic setting do not decay. There are other tools one can use in the periodic framework, such as techniques from discrete mathematics.…”

“…There are other tools one can use in the periodic framework, such as techniques from discrete mathematics. The paper of Grande, Kurianski and Staffilani [8] improved local well-posedness results in the whole-space setting; however, less results have been achieved in the periodic setting.…”