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Reflection of singularities of solutions to systems of differential equations
Author(s) -
Taylor Michael E.
Publication year - 1975
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160280403
Subject(s) - gravitational singularity , reflection (computer programming) , citation , taylor series , computer science , differential (mechanical device) , mathematics , calculus (dental) , algebra over a field , mathematical economics , library science , pure mathematics , mathematical analysis , physics , programming language , medicine , dentistry , thermodynamics
where P EPS(0) is a pseudo-differential operator of order zero. We make the assumption that P(y,x,q,t)=det(qiG, (y ,x , t ) ) is real and has simple characteristics. Then, as is well known (see [ 1 I), singularities of solutions to (1.1) propagate along the null bicharacteristic strips of p in the interior of 9. Actually, the reference does not quite apply, since a/ayG is not a pseudodifferential operator on 9 (see the appendix). Suppose ( x o , t o ) E T*( a s2) 0 and that j null-bicharacteristic strips of p pass over (xo,to). That means there a r e j real solutions q,,. . . ,q, of p ( O , ~ ~ , q , [ ~ ) = 0 . The associated bicharacteristics y, ( t) = (y (t), x ( t ) , (t), t( t)) solve the equations