By Peter E. Zhidkov
Emphasis is on questions normal of nonlinear research and qualitative thought of PDEs. fabric is said to the author's try and remove darkness from these really attention-grabbing questions no longer but coated in different monographs notwithstanding they've been the topic of released articles. Softcover.
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Extra resources for Korteweg de Vries and nonlinear Schroedinger equation
Example text
2) for be continued can C' Q0, into = 2 proof. be studied can the linear and the map yo i by analogy. with h E C ([0, problem w(O) of neighborhood a Further, interval an y(r) ) We shall consider One can - corre- then solutions 0, y(r) + 6), where 6 > 8, F0 is continuous This from (y, - 6, -go + 0, 6) = wo E has at most zero N > 3 because the case deduce easily h(r), = an r w'(0) R, one the direct verification shows that of the nonlinear the only the by solving case equation that a); R) r1-N (r N-IWI)l in b], ('go segment [0, from > E R the Vo some b [O,a).
3. E C(I; H). iwt - (-Ti, T2) = Consider the linear g(t) Dw + We understand its solutions 0, t E equation an arbitrary problem I, w(O) functions from as initial condition in Hs and the = be in the = C(I; Hs) sense wo E n H. 5) H'-2) satisfying of Hs-2 for each t E I. 5). Let it suffices to show that the us prove the unique solution w(t) Let w(-) E C(I; p(t, r) the function this solution. 6) 0. 6). Consider e-i(t-r)Dw(r) E C(I; Hs) U C'(I; H3-2) Then, we have for = a . -i(t-r)Ddw(r) dr -De-i(t-r)DW(r) + e-i(t-r)DDw(r) in the space Hs-2 I hence w(t) p(t, t) = 0 0 and thus w(r) =- 0, i.
3)) and r an JxJ) = with > w infinite sequence of (on solutions Problems with this superlinear 0 and pairwise However, In the present book literature, lutions of the there we voted to it. 3) problem. Using, for example, the estimate + aXk 0 and p > R, the derivative V(xi) sufficiently R. We recall has nontrivial solutions. Here 10WI in this direction. nonlinearities. 1). 5) nontrivial solutions. Let the maximum on shall deal with. As for the non-radial k(x1)J0JP-1, = C, and C2 independent of xi :N 22 we and Theorem important results investigations 0 for all x, E > -- under similar with p < and solution [99].