Long time decay and asymptotics for the complex mKdV equation
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913679796076544 |
|---|---|
| author | Stewart, Gavin |
| author_facet | Stewart, Gavin |
| contents | We study the asymptotics of the complex modified Korteweg-de Vries equation $\partial_t u + \partial_x^3 u = -|u|^2 \partial_x u$, which can be used to model vortex filament dynamics. In the real-valued case, it is known that solutions with small, localized initial data exhibit modified scattering for $|x| \geq t^{1/3}$ and behave self-similarly for $|x| \leq t^{1/3}$. 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 makes the low-frequency dynamics harder to control. To overcome this difficulty, we introduce 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. By using the explicit expression for $S$, we are able to get better low-frequency behavior for $u$ than we could from dispersive estimates alone. An advantage of our method is its robustness: It does not depend on the precise algebraic structure of the equation, and as such can be more readily adapted to other contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_00630 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Long time decay and asymptotics for the complex mKdV equation Stewart, Gavin Analysis of PDEs 35Q53 (Primary) 35B40 (Secondary) We study the asymptotics of the complex modified Korteweg-de Vries equation $\partial_t u + \partial_x^3 u = -|u|^2 \partial_x u$, which can be used to model vortex filament dynamics. In the real-valued case, it is known that solutions with small, localized initial data exhibit modified scattering for $|x| \geq t^{1/3}$ and behave self-similarly for $|x| \leq t^{1/3}$. 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 makes the low-frequency dynamics harder to control. To overcome this difficulty, we introduce 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. By using the explicit expression for $S$, we are able to get better low-frequency behavior for $u$ than we could from dispersive estimates alone. An advantage of our method is its robustness: It does not depend on the precise algebraic structure of the equation, and as such can be more readily adapted to other contexts. |
| title | Long time decay and asymptotics for the complex mKdV equation |
| topic | Analysis of PDEs 35Q53 (Primary) 35B40 (Secondary) |
| url | https://arxiv.org/abs/2111.00630 |