By Dr. E. Kamke

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F E C 0, ~(€1. Construction o f solutions f o r ( P I - ) L e t us c o n s i d e r t h e problem ( P I - ) and i n t r o d u c e t h e new independent variable I, E, = x / s and s e t (6,u-,v-) 6 = t o s t r e t c h the i n t e r v a l I- onto t h e f i x e d i n t e r v a l Then, o u r problem becomes t o f i n d t h e t r i p l e t s E/S. 11, we have reduced t o t h e s c a l a r problem: v(E) 5 u , and ( 2 . 3) is 32 Hiioshi FIJ,JII and Yuzo HOSONO u ( 0 ) = a. 4) t h e r e e x i s t s a unique monotone i n c r e a s i n g s o l u t i o n only f o r We look f o r a s o l u t i o n become (U-,u) and Theorem 1.

21) i n [5]), + ft F for z, (x,y) E Q ( t ) , The v e l o c i t y p o t e n t i a l complex v e l o c i t y p o t e n t i a l Lema 4 . 1 . Also s i n c e v p < 'd p1 < po. 5). The velocity potential (x,Sy), (x,y) E Q ( t ) ,It1 @ It1 < a(pl - p), and i s the real p a r t o f the Hence we have cPlt,x,y;GI a[pg-pl, i s a harmonic function 6 E CO,~], i . e . , it s a t i s f i e s equation f l . l ) , and it i s i n f i n i t e Z y many times d i f f e r e n t i a b t e s with respect t o 6 E CO,11 w i t h values i n analytic functions of (x,y) in Water Waves and Friedrichs Expansion The d e r i v a t i v e s 0 Ox, Y and s i m i l a r expansions t o ( 4 .

2 Roughly speaking, t h e y r e p r e s e n t s g h b d (i. iation4 o f secondary b i f u r c a t e d branches, b i f u r c a t e d from p r i m a r y branches o f s o l u t i o n s w i t h c e r t a i n s p a t i a l group symmetry. The l a t t e r ones have been born as p r i m a r y b i f u r c a t e d branches from t h e t r i v i a l ( = constant s t a t e ) solutions. t h e phenomenon 06 Thus, t h e N - s o l u t i o n s a r e r e s p o n s i b l e t o a e c o v u ~ y06 b h b & L t y o f primary branches.