Long time decay and asymptotics for the complex mKdV equation

Abstract

We study the asymptotics of the complex modified Korteweg-de Vries equation ∂tu + ∂ 3x u = ±|u| 2 ∂xu In the real valued case, it is known that solutions with small, localized initial data exhibit modified scattering for |x| ≥ t 1/3 , and behave self-similarly for |x| ≤ t 1/3 . We prove that the same asymptotics hold for complex mKdV. The major difficulty in the complex case is that the nonlinearity cannot be expressed as a derivative, which prevents us from using the scaling vector field to get control in weighted L 2 spaces. Instead, we must argue carefully about how wave packets at different frequencies interact in physical space while exploiting cancellations to prevent a loss of derivatives. A key ingredient in our argument is the decomposition u = S + w, where S is a self-similar solution with the same mean as u and w is a remainder that has better decay.

On The large Time Asymptotics of Klein-Gordon type equations with General Data-I

“…See e.g. [48,10]. We expect a similar behavior of the weakly localized part of the solutions of KG equations.…”

Asymptotics for small data solutions of the Ablowitz-Ladik equation

“…However, by writing the inner product in the Fourier domain and using trigonometric identities it is possible, in essence, to move the portion of the cos multiplier that is responsible for the bilinear decay from nw n onto the |u n | 2 term (see Corollary 7), recovering the t −1 decay and letting us control this term. Although this identity might appear miraculous, it should also be expected, since a similar identity holds for complex mKdV using integration by parts (see [28]).…”

“…Proof. We will focus on proving (28): the argument for (29) is similar. Using the support assumptions on f n and h n and (26), we can write…”

“…If we instead take u n (t) = (−i) n e −2it hW (h(n + 2t), −h 3 t), a similar manipulation shows that u n (t) ≈ i n e −2it W (n + 2t, −t) for a different solution W to (3). This relation to complex mKdV will allow us to use arguments similar to those from [28] to obtain asymptotics for (1).…”

On the Existence of Self-Similar solutions for some Nonlinear Schrödinger equations

“…In [29] it is shown that for small data there is a spreading weakly localized part for the KdV type equation.…”