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- W2410291226 abstract "The Davey-Stewartson (D-S) systems are written $$begin{gathered} left{ begin{gathered} i{partial _t}u + {c_0}partial _{{x_1}}^2u + partial _{{x_2}}^2u = {c_1}{left| u right|^2}u + {c_2}u{partial _{{x_1}}}varphi ,(t,x) in R times {R^2}, hfill partial _{{x_1}}^2varphi + {c_3}partial _{{x_2}}^2varphi = {partial _{{x_1}}}{left| u right|^2}, hfill uleft( {0,x} right) = {u_0}left( x right),x in {R^2}, hfill end{gathered} right. hfill hfill end{gathered}]$$where c 0, c 3 ∈ R, c 1, c 2 ∈ C. The (D-S) systems were derived by Davey-Stewartson [8], Benney-Roskes [5] and Djordjevic-Redekopp [9] and model the evolution of weakly nonlinear water waves that travel predominantly in one direction, but in which the wave amplitude is modulated slowly in horizontal directions. Ablowitz and Haberman [2] and Cornille [7] obtained a particular form of (D-S) which is considered an example of a completely integrable model which generalizes the one-dimensional Schrodinger equation. By Djordjevic-Redekopp [9] it was shown that the parameter c3 can become negative when capillary effects are important. When( c 0 c 1, c2, c3) = (1, -1, 2, -1), (-1, -2,1,1) or (-1, 2, -1,1) the system (D-S) is referred in the inverse scattering literature as the DSI, DSII defocusing and DSII focusing, respectively. In these cases several results concerning the existence of soli-tons or lump solutions, the Cauchy problem have been established ([1], [2], [3], [11], [12]) by the inverse scattering techniques. Ghidaglia and Saut [13] classified (1) as elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the respective sign of (c0, c3): (+, +), (+, -), (-, +) and (-, -)." @default.
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- W2410291226 date "2000-01-01" @default.
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- W2410291226 title "Analytic Smoothing Effects for Nonlocal Nonlinear Shrödoinger Equations in Two Space Dimensions" @default.
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