论文标题
Zakharov-Kuznetsov类型方程的渐近K-Soliton样解决方案
Asymptotic K-soliton-like Solutions of the Zakharov-Kuznetsov type equations
论文作者
论文摘要
我们在这里研究Zakharov-Kuznetsov方程，尺寸为$ 2 $和$ 3 $，以及修改后的Zakharov-Kuznetsov方程，尺寸为$ 2 $。这些方程式承认的是孤子，其特征是它们的速度和转变。给定$ k $ silitons $ r^k $（具有不同速度）的参数，我们证明了多索顿$ u $的存在和唯一性，以便$ \ left \ | | u- \ sum_ {k = 1}^k r^k \ right \ | _ {h^1} \ rightarrow 0 $ as $ t \ rightarrow +\ infty $。收敛性发生在$ h^s $中，所有$ s \ geq 0 $的指数利率。该结构是通过多索顿的连续近似来进行的。我们使用经典论证来控制错误的$ h^1 $ norms（受Martel [21]的启发），并通过接近单调性的技术引入了一种新成分，以控制$ h^s $ norm-dimension $ d \ geq2 $。
We study here the Zakharov-Kuznetsov equation in dimension $2$ and $3$ and the modified Zakharov-Kuznetsov equation in dimension $2$. Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of $K$ solitons $R^k$ (with distinct velocities), we prove the existence and uniqueness of a multi-soliton $u$ such that $\left\| u- \sum_{k=1}^K R^k \right\|_{H^1}\rightarrow 0 $ as $t \rightarrow +\infty$. The convergence takes place in $H^s$ with an exponential rate for all $s\geq 0$. The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of $H^1$-norms of the errors (inspired by Martel [21]), and introduce a new ingredient for the control of the $H^s$-norm in dimension $d \geq2$, by a technique close to monotonicity.
