Asymptotic Behaviour for Schrodinger Equations with a Quadratic Nonlinearity in One-Space Dimension
dc.contributor.author | Hayashi, Nakao | |
dc.contributor.author | Naumkin, Pavel I. | |
dc.date.accessioned | 2020-06-30T16:53:04Z | |
dc.date.available | 2020-06-30T16:53:04Z | |
dc.date.issued | 2001-07-25 | |
dc.description.abstract | We consider the Cauchy problem for the Schrödinger equation with a quadratic nonlinearity in one space dimension iut + 1/2 uxx = t-α |ux|2, u(0, x) = u0(x), where α ∈ (0, 1). From the heuristic point of view, solutions to this problem should have a quasilinear character when α ∈ (1/2, 1). We show in this paper that the solutions do not have a quasilinear character for all α ∈ [1/2, 1) if the initial data u0 ∈ H3,0 ∩ H2,2 are small, then the solution has a slow time decay such as t-α/2. For α ∈ (0,1/2), if we assume that the initial data u0 are analytic and small, then the small time decay occurs. | |
dc.description.department | Mathematics | |
dc.format | Text | |
dc.format.extent | 18 pages | |
dc.format.medium | 1 file (.pdf) | |
dc.identifier.citation | Hayashi, N., & Naumkin, P. I. (2001). Asymptotic behaviour for Schrodinger equations with a quadratic nonlinearity in one-space dimension. Electronic Journal of Differential Equations, 2001(54), pp. 1-18. | |
dc.identifier.issn | 1072-6691 | |
dc.identifier.uri | https://hdl.handle.net/10877/11925 | |
dc.language.iso | en | |
dc.publisher | Southwest Texas State University, Department of Mathematics | |
dc.rights | Attribution 4.0 International | |
dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
dc.source | Electronic Journal of Differential Equations, 2001, San Marcos, Texas: Southwest Texas State University and University of North Texas. | |
dc.subject | Schrodinger equation | |
dc.subject | Large time behaviour | |
dc.subject | Quadratic nonlinearity | |
dc.title | Asymptotic Behaviour for Schrodinger Equations with a Quadratic Nonlinearity in One-Space Dimension | |
dc.type | Article |