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In view of the Now the function Therefore, equation, u(x, t) call the function of solution a I k 1, = on is the oo 0, depending only such that this solution > 0 (resp. T2* == k 2, g(u) be = and I be u the complex-valued generalized contin- zero. ,t) of the of time (-T, T). Further, there exist be continued can complex-valued (k and in the interval oo) if limsup JJJu(-,t)JJJA; a complex argument. Then, for +oo X' on the interval and (-T,*, T2*) (resp. 15) oo. u(-, t) in CQ-Ti, T2]; X') as n T2 a a interval containing an integer and g(-) be t--T,*+O This solution then, following.

The = into the KdVE leads to the equation R, E c approach + 01" a- 0. 1) a 0 with a = 7(0) and -wa- f f (s)ds. 1) qualitative analysis First of all, waves be solved can into the NLSE leads to by quadratures. 1) derivative 0"(-) is is equation. Of course, it is However, simpler half-interval on a of the [a, b), then it follows from this bounded, where b. 1) our a and of this solution a considering b, < equation that therefore the first derivative bounded, too, a Indeed, b is also bounded. to make continuously differentiable function a right half-neighborhood is similar consider here.

C000 is dense Qoo 00 11g1ji, = DN f rN-1 [g2 (r) + (g'(r) )2]dr 0 where the constant of the linear space C07, be a DN > 0 depends only on N. Hence the space Co7, equipped with the norm 11 111, - sequence to h in converging H,. Then Let h E obviously H,1 H,' is and for any r a completion jgn}n=1,2,3,... 2-3) n-oo and for any a, b is continuous on : 0 < a < b the half-line as gn r > 0. n -4 Further, oo in H'(a, b); in since the sequence particular jgn}n=1,2,3,... 2. EXISTENCE OF SOLUTIONS. 3) easily imply h(jxj) = almost everywhere RN in .

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