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

Half I of this quantity surveys the advancements within the research of nonlinear phenomena in Japan in the past decade, whereas half II includes up to date unique papers touching on qualitative theories and their functions. handled listed here are nonlinear difficulties on the topic of normal research, fluid dynamics, mathematical biology and laptop sciences, and their underlying mathematical constructions, e.g. nonlinear waves and propagations, bifurcation phenomena, chaotic phenomena, and fractals. the quantity is devoted to Professor Masaya Yamaguti in occasion of his sixtieth birthday.

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**Extra info for Patterns and Waves: Qualitative Analysis of Nonlinear Differential Equations**

**Sample text**

We shall start from the Monge-Ampere equation as well as R. Courant and D. Hilbert, Appendix 1 to Chapter V [4]. 1. The real Monge-AmpZre equation This is one of the typical nonlinear equations closely related to geometry; It appears for examples in the local construction of a piece of hypersurface with a given Gauss curvature, the local embedding on R3 of a 2-dimensional manifold with given a smooth Riemannian metric, and also a similar problem of locally connectable isometric surfaces by a bending (the problem of E.

3), we get (a,+ V$* V $ -f'"($)d$ =g (j)= . B)d$ = -f ( ' ) d $-f ( 2 )I V$ 12- 18%)12 . Then we obtain two equations (a,+ B$ * V ) 2 $ - - f'"d$- (f'1 +f (2') IV$I'IP"l~ =o and (a,+ V$* V $ -f(1Y$)4 =g It is clear that we can attain the conclusion independent of g if we find a solution of this system under a n initial condition satisfying on a n initial surface . + a =(a,+V$ V)$h A$ =0 because these equations mean and + + (8, V$ * V ) 6 Vz$ * 6- F ( f (')($)u)=0 , and because the uniqueness of this linear system is also assured.

Pure Appl. , 7 (1954), 354-392. , 1968. L. , 1958. , 85 (1950), 1-62. J. Hadamard, Le Probltme de Cauchy et les Equations aux Dtrivtes Partielles Lintaires Hyperboliques, Herrnann, Paris, 1932. L. Horrnander, The boundary problems of physical geodesy, Arch, Rational Mech. , 62 (1976), 1-52. -, The Cauchy problem for differential equations with double characteristics, J. , 32 (1977), 118-196. Strongly Hyperbolic Equations 35 [13] V. Ya. Ivrii and V. M. Petkov, Necessary conditions for the Cauchy problem for nonstrictly hyperbolic equations to be well-posed, Uspehi Mat.