论文标题
在所有维度上的分数广义KDV方程中孤立波的稳定性和不稳定
Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions
论文作者
论文摘要
我们研究了分数通用的korteweg-de vries方程$$ \ partial_t u- \ partial_ {x_1} d^αu+ \ tfrac {1} {m} {m} {m} {m} \ Mathbb {r}^d，\，\，t \ in \ mathbb {r}，\，\，\，0 <α<2，$ $ in任何空间维度$ d \ geq 1 $和非线性$ m> 1 $。这里提出的论点独立于空间维度，并依靠新的估计值及其规律性的空间衰减。在$ l^2 $ - 简单的情况下，我们使用浓度 - 触觉参数证明了孤立波的轨道稳定性，在几个变量中，非本地操作员$ d^α$的换向器估计和扩展。在$ l^2 $ -Supergitical的情况下，我们表明孤独的波浪是不稳定的。更确切地说，通过在有限的时间内构建明确的初始条件序列，从孤子轨道上移开，这与调制和截断参数结合在一起，并结合了接地状态的衰减和规律性，从而获得了不稳定性。 结果，在1D中，我们显示了超批评性概括的本杰明·诺克方程（$α= 1 $）的孤立波和分散剂的benjamin-ono方程（$ 1 <α<2 $）；此外，当$ \ frac {1} {2} {2} <α<1 $ $时，在较弱的分散体制中获得了有关不稳定的新结果。这项工作应该引起研究，以研究涉及非局部操作员的各种分散方程中孤立波和其他相干结构的稳定性。
We study stability of solitary wave solutions for the fractional generalized Korteweg-de Vries equation $$ \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.
