By Peter E. Zhidkov
Emphasis is on questions standard of nonlinear research and qualitative conception of PDEs. fabric is said to the author's try to light up these fairly attention-grabbing questions no longer but coated in different monographs although they've been the topic of released articles. Softcover.
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Extra info for Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory
5) is proved. , t) 0 thus of the E dr w(r) problem C(I; H') (we =- 0, i. e. 2). remind that we of the linear as in the case 1). 4). 4). 2. 2). 2. 4. EQUATION(NLSE) above, the by paper T. Kato presented. is we followed 29 this prove only theorem for N = 1. 00 Also, f(JU12 )u that we assume U C2((_rn,,rn) E x (-n, to the n); R2) for simplicity. , t) and ul (*) t) U2 equivalent is We first equation. and by embedding (1. 2). problem So, solution. Let C(I; H'). 4), = of For t < 0 this t > 0. solution a the an of T existence for of prove us f(JU12)U that (*, t) _= U2 uniqueness (Gu)(t) = solution.
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Then, all it r we > 1) the an for solution where Cauchy problem this we follow I > 0 is exactly has the y(r) of . 8) 0, half-line entire > r r addition, In > 0. 4), theorem large 2Co(a2 > that equation [0, ro]. E a-2)(l + )2 taking on > 0 ro from Let ri and r2 (11. 1. 8), (11. 1. 1*2 2(N we + yo from continuous such 2)(N G(y(r)) the the energy interval dependence that that - 1) ro follows - step, take roots, solution any on sufficiently a with of which 1) + (a,, a2), Y/2. function of and E 1 solution G(y(ro)) and : > 0 the to With (0, a2) y E y(r) We call a close E yo show that we estimate Co exists solutions.