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Existence of two-solitary waves with logarithmic distance for the nonlinear Klein–Gordon equation
Communications in Contemporary Mathematics ( IF 1.6 ) Pub Date : 2020-12-19 , DOI: 10.1142/s0219199720500911
Shrey Aryan 1
Affiliation  

We consider the focusing nonlinear Klein–Gordon (NLKG) equation ttu Δu + u |u|p1u = 0, (t,x) × d for 1 d 5 and p > 2 subcritical for the 1 norm. In this paper, we show the existence of a solution u(t) of the equation such that u(t) k=1,2Qk(t)H1 + tu(t)L2 0as t +, where Qk(t,x) are two solitary waves of the equation with translations zk : d satisfying |z1(t) z2(t)| 2log(t)as t +. This behavior is due to the strong interactions between solitary waves which is in contrast with the previous work [R. Côte and Y. Martel, Multi-travelling waves for the nonlinear Klein–Gordon equation, Trans. Amer. Math. Soc. 370(10) (2018) 7461–7487] on multi-solitary waves of the (NLKG), devoted to the case of solitary waves with different speeds. This work is motivated by previous similar existence results for the nonlinear Schrödinger and generalized Korteweg–de Vries equations.

中文翻译:

非线性 Klein-Gordon 方程存在对数距离的两个孤立波

我们考虑聚焦非线性 Klein-Gordon (NLKG) 方程 - Δ + -||p-1 = 0, (,X) × d 为了1 d 5p > 2亚临界的H1规范。在本文中,我们证明了解决方案的存在()的等式使得 () -ķ=1,2ķ()H1 + ()大号2 0作为  +, 在哪里ķ(,X)是具有平移的方程的两个孤立波zķ d令人满意的 |z1() - z2()| 2日志()作为  +. 这种行为是由于孤立波之间的强相互作用,这与之前的工作 [R. Côte 和 Y. Martel,非线性 Klein-Gordon 方程的多行波,反式。阿米尔。数学。社会党。 370(10) (2018) 7461-7487] 关于(NLKG)的多孤波,专门研究不同速度的孤波的情况。这项工作的动机是先前非线性薛定谔方程和广义 Korteweg-de Vries 方程的类似存在结果。
更新日期:2020-12-19
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