Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data
We study the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with a step-like initial data: $q(x,0)=o(1)$ as $x\to-\infty$ and $q(x,0)=A+o(1)$ as $x\to\infty$, where $A>0$ is an arbitrary constant. W...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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V.N. Karazin Kharkiv National University Publishing
2018-12-01
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| Series: | Visnik Harkivsʹkogo Nacionalʹnogo Universitetu im. V.N. Karazina. Cepiâ Matematika, Prikladna Matematika i Mehanika |
| Subjects: | |
| Online Access: | https://periodicals.karazin.ua/mech_math/article/view/12134 |
| _version_ | 1831790401615298560 |
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| author | Ya. Rybalko D. G. Shepelsky |
| author_facet | Ya. Rybalko D. G. Shepelsky |
| author_sort | Ya. Rybalko |
| collection | DOAJ |
| description | We study the Cauchy problem for
the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation
\[
iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0
\]
with a step-like initial data: $q(x,0)=o(1)$ as $x\to-\infty$ and $q(x,0)=A+o(1)$ as $x\to\infty$, where $A>0$ is an arbitrary constant. We develop the inverse scattering transform method for this problem in the
form of the Riemann-Hilbert approach and obtain the representation of the solution of the Cauchy
problem in terms of the solution of an associated Riemann-Hilbert-type analytic factorization problem,
which can be efficiently used for further studying the properties of the solution, including the large time
asymptotic behavior. |
| first_indexed | 2024-12-22T14:31:34Z |
| format | Article |
| id | doaj.art-ba4d87577a2845d8873d2a1dfd8e84c7 |
| institution | Directory Open Access Journal |
| issn | 2221-5646 2523-4641 |
| language | English |
| last_indexed | 2024-12-22T14:31:34Z |
| publishDate | 2018-12-01 |
| publisher | V.N. Karazin Kharkiv National University Publishing |
| record_format | Article |
| series | Visnik Harkivsʹkogo Nacionalʹnogo Universitetu im. V.N. Karazina. Cepiâ Matematika, Prikladna Matematika i Mehanika |
| spelling | doaj.art-ba4d87577a2845d8873d2a1dfd8e84c72022-12-21T18:22:44ZengV.N. Karazin Kharkiv National University PublishingVisnik Harkivsʹkogo Nacionalʹnogo Universitetu im. V.N. Karazina. Cepiâ Matematika, Prikladna Matematika i Mehanika2221-56462523-46412018-12-018841612134Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial dataYa. Rybalko0D. G. Shepelsky1B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of UkraineB.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of UkraineWe study the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with a step-like initial data: $q(x,0)=o(1)$ as $x\to-\infty$ and $q(x,0)=A+o(1)$ as $x\to\infty$, where $A>0$ is an arbitrary constant. We develop the inverse scattering transform method for this problem in the form of the Riemann-Hilbert approach and obtain the representation of the solution of the Cauchy problem in terms of the solution of an associated Riemann-Hilbert-type analytic factorization problem, which can be efficiently used for further studying the properties of the solution, including the large time asymptotic behavior.https://periodicals.karazin.ua/mech_math/article/view/12134nonlocal nonlinear Schr\ |
| spellingShingle | Ya. Rybalko D. G. Shepelsky Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data Visnik Harkivsʹkogo Nacionalʹnogo Universitetu im. V.N. Karazina. Cepiâ Matematika, Prikladna Matematika i Mehanika nonlocal nonlinear Schr\ |
| title | Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data |
| title_full | Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data |
| title_fullStr | Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data |
| title_full_unstemmed | Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data |
| title_short | Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data |
| title_sort | riemann hilbert approach for the integrable nonlocal nonlinear schr odinger equation with step like initial data |
| topic | nonlocal nonlinear Schr\ |
| url | https://periodicals.karazin.ua/mech_math/article/view/12134 |
| work_keys_str_mv | AT yarybalko riemannhilbertapproachfortheintegrablenonlocalnonlinearschrodingerequationwithsteplikeinitialdata AT dgshepelsky riemannhilbertapproachfortheintegrablenonlocalnonlinearschrodingerequationwithsteplikeinitialdata |