The inverse scattering transform for multidimensional (2+1) problems

Page 1 In t roduc t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 1.1 F ind a m e t h o d for genera t ing par t icu la r solut ions of t h e given equa t ion . . . . . . 140 1.2 F ind a ' m e t h o d for solving general initial value p rob lems . . . . . . . . . . . . . . . 140 2 Direct l inear izat ion in 1 + 1 and 2 + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3 Review of t he ma i n eigenvalue p rob lems re la ted to t he inverse sca t t e r ing t r a n s f o r m in 1 + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4 R e m a r k s on R i e m a n n H i l b e r t b o u n d a r y value problems . . . . . . . . . . . . . . . . . . 146 4.1 Classical R i e m a n n H i i b e r t p roblems for HSlder func t ions . . . . . . . . . . . . . . 146 4.1.1 Scalar homogeneous R i e m a n n H i l b e r t p roblem . . . . . . . . . . . . . . . . . 147 4.1.2 Vector homogeneous R i e m a n n H i l b e r t p rob lems . . . . . . . . . . . . . . . . 148 4.1.3 A note on the h is tory of R i e m a n n H i l b e r t p rob lems . . . . . . . . . . . . . . 150 4.2 Some resul ts of Kre in and Gohberg . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2.1 Scalar fac tor iza t ion prob lem . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.2.2 A vector R i e m a n n H i l b e r t p rob lem wi th all i ts indices posit ive . . . . . . . . 152 4.2.3 A ma t r i x fac tor iza t ion t heo rem . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2.4 A the o rem abou t indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.3 R i e m a n n H i l b e r t p roblems appear ing in inverse sca t t e r ing t r a n s f o r m s . . . . . . . 154

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