Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions
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arXiv
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| Format: | Preprint |
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2022
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| author | Riaño, Oscar Roudenko, Svetlana |
| author_facet | Riaño, Oscar Roudenko, Svetlana |
| contents | We study stability of solitary wave solutions for the fractional generalized Korteweg-de Vries equation $$ \partial_t u- \partial_{x_1} D^αu+ \tfrac{1}{m}\partial_{x_1}(u^m)=0, ~ (x_1,\dots,x_d)\in \mathbb{R}^d, \, \, t\in \mathbb{R}, \, \, 0<α<2, $$ in any spatial dimension $d\geq 1$ and nonlinearity $m>1$. The arguments developed here are independent of the spatial dimension and rely on the new estimates for spatial decay of ground states and their regularity. In the $L^2$-subcritical case, we prove the orbital stability of solitary waves using the concentration-compactness argument, the commutator estimates and expansions of nonlocal operator $D^α$ in several variables. In the $L^2$-supercritical case, we show that solitary waves are unstable. More precisely, the instability is obtained by constructing an explicit sequence of initial conditions that move away from a soliton orbit in finite time, this is shown in conjunction with the modulation and truncation arguments, and incorporating the decay and regularity of the ground states.
As a consequence, in 1D we show the instability of solitary waves of the supercritical generalized Benjamin-Ono equation ($α=1$) and the dispersion-generalized Benjamin-Ono equation ($1<α<2$); furthermore, new results on the instability are obtained in the weaker dispersion regime when $\frac{1}{2}<α<1$. This work should be of interest in studying stability of solitary waves and other coherent structures in a variety of dispersive equations that involve nonlocal operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_09159 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions Riaño, Oscar Roudenko, Svetlana Analysis of PDEs 35B35, 35Q35, 35Q51, 35Q53 We study stability of solitary wave solutions for the fractional generalized Korteweg-de Vries equation $$ \partial_t u- \partial_{x_1} D^αu+ \tfrac{1}{m}\partial_{x_1}(u^m)=0, ~ (x_1,\dots,x_d)\in \mathbb{R}^d, \, \, t\in \mathbb{R}, \, \, 0<α<2, $$ in any spatial dimension $d\geq 1$ and nonlinearity $m>1$. The arguments developed here are independent of the spatial dimension and rely on the new estimates for spatial decay of ground states and their regularity. In the $L^2$-subcritical case, we prove the orbital stability of solitary waves using the concentration-compactness argument, the commutator estimates and expansions of nonlocal operator $D^α$ in several variables. In the $L^2$-supercritical case, we show that solitary waves are unstable. More precisely, the instability is obtained by constructing an explicit sequence of initial conditions that move away from a soliton orbit in finite time, this is shown in conjunction with the modulation and truncation arguments, and incorporating the decay and regularity of the ground states. As a consequence, in 1D we show the instability of solitary waves of the supercritical generalized Benjamin-Ono equation ($α=1$) and the dispersion-generalized Benjamin-Ono equation ($1<α<2$); furthermore, new results on the instability are obtained in the weaker dispersion regime when $\frac{1}{2}<α<1$. This work should be of interest in studying stability of solitary waves and other coherent structures in a variety of dispersive equations that involve nonlocal operators. |
| title | Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions |
| topic | Analysis of PDEs 35B35, 35Q35, 35Q51, 35Q53 |
| url | https://arxiv.org/abs/2210.09159 |