Riemann-Hilbert approach for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data
Keyword(s):
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.
2017 ◽
Vol 108
(1)
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pp. 41-62
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2019 ◽
Vol 07
(10)
◽
pp. 2333-2351
2020 ◽
Vol 10
(1)
◽
pp. 353-370
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Keyword(s):
2021 ◽
Vol 18
(03)
◽
pp. 701-728