Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces
We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces i∂tu+(1/2)Δu=𝒩(u), (t,x)∈ℝ×ℝ2;u(0,x)=φ(x), x∈ℝ2, where 𝒩(u)=Σj,k=12(λjk(∂xju)(∂xku)+μjk(∂xju¯)(∂xku¯)), where λjk,μjk∈ℂ. We prove that if t...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202007652 |
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| author | Nakao Hayashi Pavel I. Naumkin |
| author_facet | Nakao Hayashi Pavel I. Naumkin |
| author_sort | Nakao Hayashi |
| collection | DOAJ |
| description | We study asymptotic behavior in time of global small solutions to
the quadratic nonlinear Schrödinger equation in
two-dimensional spaces i∂tu+(1/2)Δu=𝒩(u), (t,x)∈ℝ×ℝ2;u(0,x)=φ(x), x∈ℝ2, where 𝒩(u)=Σj,k=12(λjk(∂xju)(∂xku)+μjk(∂xju¯)(∂xku¯)), where
λjk,μjk∈ℂ. We prove that if the initial data
φ satisfy some analyticity and smallness conditions in a
suitable norm, then the solution of the above Cauchy problem has
the asymptotic representation in the neighborhood of the
scattering states. |
| format | Article |
| id | doaj-art-b946e3f6d5f647b4993456bffbbc681c |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-b946e3f6d5f647b4993456bffbbc681c2025-02-03T01:28:05ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0129950151610.1155/S0161171202007652Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spacesNakao Hayashi0Pavel I. Naumkin1Department of Mathematics, Graduate School of Science, Osaka University, Osaka 560-0043, JapanInstituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia, Michoacán CP 58089, MexicoWe study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spaces i∂tu+(1/2)Δu=𝒩(u), (t,x)∈ℝ×ℝ2;u(0,x)=φ(x), x∈ℝ2, where 𝒩(u)=Σj,k=12(λjk(∂xju)(∂xku)+μjk(∂xju¯)(∂xku¯)), where λjk,μjk∈ℂ. We prove that if the initial data φ satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the above Cauchy problem has the asymptotic representation in the neighborhood of the scattering states.http://dx.doi.org/10.1155/S0161171202007652 |
| spellingShingle | Nakao Hayashi Pavel I. Naumkin Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces International Journal of Mathematics and Mathematical Sciences |
| title | Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces |
| title_full | Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces |
| title_fullStr | Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces |
| title_full_unstemmed | Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces |
| title_short | Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces |
| title_sort | asymptotic expansion of small analytic solutions to the quadratic nonlinear schrodinger equations in two dimensional spaces |
| url | http://dx.doi.org/10.1155/S0161171202007652 |
| work_keys_str_mv | AT nakaohayashi asymptoticexpansionofsmallanalyticsolutionstothequadraticnonlinearschrodingerequationsintwodimensionalspaces AT pavelinaumkin asymptoticexpansionofsmallanalyticsolutionstothequadraticnonlinearschrodingerequationsintwodimensionalspaces |