Riemann-Hilbert approach for the integrable nonlocal nonlinear Schr\"odinger equation with step-like initial data
Abstract
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.
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References
for Nonlinear Problems, Stud. Appl. Math., -- 1974. -- 53. -- P. 249--315.
M. J. Ablowitz and Z. H. Musslimani, Integrable nonlocal nonlinear Schr\"odinger equation, Phys. Rev. Lett., -- 2013. -- 110. -- P. 064105-1--064105-5.
M. J. Ablowitz and Z. H. Musslimani, Inverse scattering transform for the integrable nonlocal nonlinear Schr\"odinger equation, Nonlinearity. -- 2016. -- 29. -- P. 915--946.
C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having P-T symmetry, Phys. Rev. Lett., -- 1998. -- 80. -- P. 5243--5246.
R. F. Bikbaev, Diffraction in Nonlinear Defocusing Medium, Zapiski Nauch. Semin. LOMI. -- 1989. -- 179. -- P. 23--31.
A. Boutet de Monvel, V. P. Kotlyarov and D. Shepelsky, Focusing NLS Equation: Long-Time Dynamics of Step-Like
Initial Data, International Mathematics Research Notices. -- 2011. -- 7. -- P. 1613--1653.
P. A. Deift, A. R. Its and X. Zhou, Long-time asymptotics for integrable nonlinear wave equations,
Important developments in Soliton Theory 1980-1990, edited by A. S. Fokas and V. E. Zakharov, New York: Springer. -- 1993. -- P. 181--204.
P. A. Deift and X. Zhou, A steepest descend method for oscillatory Riemann--Hilbert problems. Asymptotics for the MKdV equation, Ann. Math. -- 1993. -- 137. -- P. 295--36
T. Gadzhimuradov and A. Agalarov, Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schr\"odinger equation, Phys. Rev. A. -- 2016. -- 93. -- P. 062124-1--062124-6.
Ya. Rybalko, D. Shepelsky, Long-time asymptotics for the integrable nonlocal nonlinear Schr\"odinger equation, {arXiv:1710.07961}, 18 Apr 2018.
A. Sarma, M. Miri, Z. Musslimani, D. Christodoulides, Continuous and discrete Schr\"odinger systems with parity-time-symmetric nonlinearities, Physical Review E. -- 2014. -- 89. -- P. 052918-1--052918-7
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