By Takaaki Nishida, etc., M. Mimura, H. Fujii

Half I of this quantity surveys the advancements within the research of nonlinear phenomena in Japan up to now decade, whereas half II comprises updated unique papers bearing on qualitative theories and their functions. handled listed below are nonlinear difficulties concerning basic research, fluid dynamics, mathematical biology and desktop sciences, and their underlying mathematical buildings, e.g. nonlinear waves and propagations, bifurcation phenomena, chaotic phenomena, and fractals. the amount is devoted to Professor Masaya Yamaguti in occasion of his sixtieth birthday.

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**Additional resources for Patterns and waves: Qualitative analysis of nonlinear differential equations**

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39) and in a closed form by u(x; t) D 2c 1Ce c (x c t) Á; iÁ ct) : (41) Camassa and Holm [4] derived in 1993 a completely integrable wave equation t4 ; and so on. The series solution is thus given by 2ce h c (x 2 Peakons of the Camassa–Holm Equation C 11e cx tanh Figure 2 shows a kink graph. The graph shows that the kink converges to ˙1 as ! ˙1. t3 ; )4 u(x; t) D c 1 ¤0; (40) u x x t C 3uu x D 2u x u x x C uu x x x (42) by retaining two terms that are usually neglected in the small-amplitude shallow-water limit [3].

Such scaling holds e. g. for the mean time spent by a walk in a prescribed region of space (i. e. for the mean ﬁrst passage time to its boundary). For an isotropic system Eq. (2) can be rewritten as an equation in the radial coordinate only: Ä @ 1 @ d 1@ P(r; t) D K d 1 r P(r; t) : (3) @t @r @r r Looking at the trajectory of a random walk at scales much larger than the step’s length one infers that it is self-similar in statistical sense, i. e. that its whatever portion (of the size considerably larger then the step’s length) looks like the whole trajectory, i.

Compactons of the K(n,n) Equation The K(n,n) equation [16,17] was introduced by Rosenau and Hyman in 1993. This equation was investigated experimentally and analytically. The K(m,n) equation is a genuinely nonlinear dispersive equation, a special type of the n>1: (59) Compactons, which are solitons with compact support or strict localization of solitary waves, have been investigated thoroughly in the literature. The delicate interaction between the eﬀect of the genuine nonlinear convection (u n )x and the genuinely nonlinear dispersion of (u n )x x x generates solitary waves with exact compact support that are called compactons.