美金汇兑人民币汇率题目：Small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting 美金汇兑人民币汇率专家：Chongchun Zeng（曾崇纯） 美国佐治亚理工学院数学系 教授 美金汇兑人民币汇率地点：南1-120教室 美金汇兑人民币汇率时间：2024年06月14日（周五）上午10:00-11:30 美金汇兑人民币汇率摘要：Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive partial differential equations (PDEs). Families of breathers have been found for certain integrable PDEs, such as the sine-Gordon equation in 1-dim, but are believed to be rare in non-integrable ones such as nonlinear Klein-Gordon equations. In this talk we consider small breathers for semilinear Klein-Gordon equations with analytic odd nonlinearities. A breather with small amplitude exists only when its temporal frequency is close to be resonant with the linear Klein-Gordon dispersion relation. Our main result is that, for such frequencies, we rigorously identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of small breathers in terms of the so-called Stokes constant which depends on the nonlinearity analytically, but is independent of the frequency. As a corollary it proves that, for generic analytic odd nonlinearities, there does not exist any small breather of any temporal frequency, even though this had been intuitive for any single given frequency due to the dimension counting. In particular, this gives a rigorous justification of a formal asymptotic argument by Kruskal and Segur (1987) in the analysis of small breathers. The proof is carried out in a singular perturbation setting in the spatial dynamics framework. The leading order term in the exponentially small splitting is obtained through a careful analysis of the analytic continuation of the parameterizations of two specials small solutions which decay weakly at the spatial infinities. This is a joint work with O. Gomide, M. Guardia, and T. Seara. 美金汇兑人民币汇率人简介：曾崇纯，美国佐治亚理工学院教授，美国数学会会士。主要从事微分方程与动力系统研究，曾获得美国Career奖和Sloan基金，并多次主持美国国家自然科学基金，在国际顶级的四大学数学期刊之一《 Inventiones Mathematicae》及《Comm. Pure Appl. Math》、《 SIAM J. Math. Anal》等国际顶级学术刊物发表论文多篇，现任《Journal of Differential Equation》和《Discrete Continuous Dynamical Systems》等杂志编委。有关曾崇纯教授的详细简历可参见链接: http://www.math.gatech.edu/~zengch
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