The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrödinger (NLS) equation with non-zero boundary values

$q_{l/r}(t)\equiv A_{l/r} e^{-2iA_{l/r}^2t+i\theta _{l/r}}$
ql/r(t)≡Al/re−2iAl/r2t+iθl/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl ≠ θr. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that
$\left(q(x,t)-q_{l/r}(t)\right)\in L^{1,1}(\mathbb {R^\mp })$
q(x,t)−ql/r(t)∈L1,1(R∓) with respect to x for all t ⩾ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables
$\lambda _{l/r}=\sqrt{k^2+A^2_{l/r}}$
λl/r=k2+Al/r2⁠, where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.

Topics
Wave propagation, Nonlinear Schrodinger equation, Integral equations, Asymptotic analysis, Gradient method, Complex analysis, Atmospheric optical phenomena, Inverse scattering, Marchenko equations, Scattering problem
1.
M. J.
 
Ablowitz
 and
D. E.
 
Baldwin
, “
Dispersive shock wave interactions and asymptotics
,”
Phys. Rev. E
 
87
(
2
),
022906
(
2013
).
https://doi.org/10.1103/PhysRevE.87.022906
Google Scholar
Crossref
Search ADS
 
2.
M. J.
 
Ablowitz
,
D. J.
 
Kaup
,
A. C.
 
Newell
, and
H.
 
Segur
, “
The Inverse scattering transform. Fourier analysis for nonlinear problems
,”
Stud. Appl. Math.
 
53
,
249
–
315
(
1974
).
Google Scholar
Crossref
Search ADS
 
3.
M. J.
 
Ablowitz
,
B.
 
Prinari
, and
A. D.
 
Trubatch
,
Discrete and Continuous Nonlinear Schrödinger Systems
,
London Mathematical Society Lecture Notes Series
Vol.
302
(
Cambridge University Press
,
Cambridge
,
2004
).
Google Scholar
 
4.
M. J.
 
Ablowitz
 and
H.
 
Segur
,
Solitons and the Inverse Scattering Transform
(
SIAM
,
Philadelphia
,
1981
).
Google Scholar
Crossref
Search ADS
 
5.
M. J.
 
Ablowitz
 and
H.
 
Segur
, “
Asymptotic solutions of the Korteweg-de Vries equation
,”
Stud. App. Math.
 
57
,
13
–
44
(
1977
).
Google Scholar
Crossref
Search ADS
 
6.
N. N.
 
Akhmediev
,
V. M.
 
Eleonskii
, and
N. E.
 
Kulagin
, “
Generation of a periodic sequence of picosecond pulses in an optical fiber. Exact solutions
,”
Sov. Phys. JETP
 
89
,
1542
–
1551
(
1985
) [in Russian].
Google Scholar
 
7.
N. N.
 
Akhmediev
,
A.
 
Ankiewicz
, and
M.
 
Taki
, “
Waves that appear from nowhere and disappear without a trace
,”
Phys. Lett. A
 
373
,
675
–
678
(
2009
).
https://doi.org/10.1016/j.physleta.2008.12.036
Google Scholar
Crossref
Search ADS
 
8.
N. N.
 
Akhmediev
,
A.
 
Ankiewicz
, and
J. M.
 
Soto-Crespo
, “
Rogue waves and rational solutions of the nonlinear Schrödinger equation
,”
Phys. Rev. E
 
80
,
026601
(
2009
).
https://doi.org/10.1103/PhysRevE.80.026601
Google Scholar
Crossref
Search ADS
 
9.
N. N.
 
Akhmediev
 and
V. I.
 
Korneev
, “
Modulational instability and periodic solutions of the nonlinear Schrödinger equation
,”
Theor. Math. Phys.
 
69
,
1089
–
1093
(
1986
).
https://doi.org/10.1007/BF01037866
Google Scholar
Crossref
Search ADS
 
10.
T. B.
 
Benjamin
, “
Instability of periodic wavetrains in nonlinear dispersive systems
,”
Proc. Roy. Soc. A
 
299
,
59
–
75
(
1967
).
https://doi.org/10.1098/rspa.1967.0123
Google Scholar
Crossref
Search ADS
 
11.
T. B.
 
Benjamin
 and
J. E.
 
Feir
, “
The disintegration of wavetrains in deep water. Part I
,”
J. Fluid Mech.
 
27
,
417
–
430
(
1967
).
https://doi.org/10.1017/S002211206700045X
Google Scholar
Crossref
Search ADS
 
12.
G.
 
Biondini
 and
G.
 
Kovačić
, “
Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions
,”
J. Math. Phys.
 
55
,
031506
(
2014
).
https://doi.org/10.1063/1.4868483
Google Scholar
Crossref
Search ADS
 
13.
A.
 
Boutet de Monvel
,
V. P.
 
Kotlyarov
, and
D.
 
Shepelsky
, “
Focusing NLS equation: Long-time dynamics of step-like initial data
,”
Int. Math. Res. Notices
 
7
,
1613
–
1653
(
2011
).
https://doi.org/10.1093/imrn/rnq129
Google Scholar
Crossref
Search ADS
 
14.
R.
 
Buckingham
 and
S.
 
Venakides
, “
Long-time asymptotics of the nonlinear Schrödinger equation shock problem
,”
Commun. Pure App. Math.
 
60
,
1349
–
1414
(
2007
).
https://doi.org/10.1002/cpa.20179
Google Scholar
Crossref
Search ADS
 
15.
F.
 
Calogero
 and
A.
 
Degasperis
,
Spectral Transforms and Solitons
(
North-Holland
,
Amsterdam
,
1982
).
Google Scholar
 
16.
M.
 
Chen
,
M. A.
 
Tsankov
,
J. M.
 
Nash
, and
C. E.
 
Patton
, “
Backward-volume-wave microwave-envelope solitons in yttrium iron garnet films
,”
Phys. Rev. B
 
49
,
12773
–
12790
(
1994
).
https://doi.org/10.1103/PhysRevB.49.12773
Google Scholar
Crossref
Search ADS
 
17.
P.
 
Deift
 and
X.
 
Zhou
, “
A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation
,”
Ann. Math.
 
137
,
295
–
368
(
1993
).
https://doi.org/10.2307/2946540
Google Scholar
Crossref
Search ADS
 
18.
F.
 
Demontis
,
B.
 
Prinari
,
C.
 
van der Mee
, and
F.
 
Vitale
, “
The inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonzero boundary conditions
,”
Stud. Appl. Math.
 
131
,
1
–
40
(
2013
).
https://doi.org/10.1111/j.1467-9590.2012.00572.x
Google Scholar
Crossref
Search ADS
 
19.
L. D.
 
Faddeev
 and
L. A.
 
Takhtajan
,
Hamiltonian Methods in the Theory of Solitons
(
Springer
,
Berlin
,
1987
).
Google Scholar
Crossref
Search ADS
 
20.
J.
 
Garnier
 and
K.
 
Kalimeris
, “
Inverse scattering perturbation theory for the nonlinear Schrödinger equation with non-vanishing background
,”
J. Phys. A: Math. Theor.
 
45
,
035202
(13pp.) (
2012
).
https://doi.org/10.1088/1751-8113/45/3/035202
Google Scholar
Crossref
Search ADS
 
21.
A.
 
Hasegawa
, and
F.
 
Tappert
, “
Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion and II. Normal dispersion
,”
Appl. Phys. Lett.
 
23
(
3
),
142
–
144
and
–
171
(
1973
).
https://doi.org/10.1063/1.1654836
Google Scholar
Crossref
Search ADS
 
22.
A. R.
 
Its
,
A. V.
 
Rybin
, and
M. A.
 
Sall
, “
Exact integration of nonlinear Schrödinger equation
,”
Theor. Math. Phys.
 
74
,
20
–
32
(
1988
).
https://doi.org/10.1007/BF01018207
Google Scholar
Crossref
Search ADS
 
23.
E. A.
 
Kuznetsov
, “
Solitons in a parametrically unstable plasma
,”
Sov. Phys. Dokl. (Engl. Transl.)
 
22
,
507
–
508
(
1977
).
Google Scholar
 
24.
Y.-C.
 
Ma
, “
The perturbed plane-wave solutions of the cubic Schrödinger equation
,”
Stud. Appl. Math.
 
60
,
43
–
58
(
1979
).
Google Scholar
Crossref
Search ADS
 
25.
D.
 
Mihalache
,
F.
 
Lederer
, and
D-M
 
Baboiu
, “
Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical soliton propagation
,”
Phys. Rev. A
 
47
,
3285
–
3290
(
1993
).
https://doi.org/10.1103/PhysRevA.47.3285
Google Scholar
Crossref
Search ADS
PubMed
 
26.
S. P.
 
Novikov
,
S. V.
 
Manakov
,
L. B.
 
Pitaevskii
, and
V. E.
 
Zakharov
,
Theory of Solitons. The Inverse Scattering Method
(
Plenum Press
,
New York
,
1984
).
Google Scholar
 
27.
Y.
 
Ohta
 and
J.
 
Yang
, “
General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation
,”
Proc. Roy. Soc. A
 
468
,
1716
–
1740
(
2012
).
https://doi.org/10.1098/rspa.2011.0640
Google Scholar
Crossref
Search ADS
 
28.
M.
 
Onorato
,
A. R.
 
Osborne
, and
M.
 
Serio
, “
Modulational instability in crossing sea states: A possible mechanism for the formation of freak waves
,”
Phys. Rev. Lett.
 
96
,
014503
(
2006
).
https://doi.org/10.1103/PhysRevLett.96.014503
Google Scholar
Crossref
Search ADS
PubMed
 
29.
C. J.
 
Pethick
 and
H.
 
Smith
,
Bose-Einstein Condensation in Dilute Gases
(
Cambridge University Press
,
Cambridge
,
2002
).
Google Scholar
 
30.
D. H.
 
Peregrine
, “
Water waves, nonlinear Schrödinger equations and their solutions
,”
J. Aust. Math. Soc. Ser. B
 
25
,
16
–
43
(
1983
).
https://doi.org/10.1017/S0334270000003891
Google Scholar
Crossref
Search ADS
 
31.
D. R.
 
Solli
,
C.
 
Ropers
,
P.
 
Koonath
, and
B.
 
Jalali
, “
Optical rogue waves
,”
Nature (London)
 
450
,
1054
–
1057
(
2007
).
https://doi.org/10.1038/nature06402
Google Scholar
Crossref
Search ADS
 
32.
M.
 
Tajiri
 and
Y.
 
Watanabe
, “
Breather solutions to the focusing nonlinear Schrödinger equation
,”
Phys. Rev. E
 
57
,
3510
–
3519
(
1998
).
https://doi.org/10.1103/PhysRevE.57.3510
Google Scholar
Crossref
Search ADS
 
33.
V. E.
 
Zakharov
, “
Hamilton formalism for hydrodynamic plasma models
,”
Sov Phys. JETP
 
33
,
927
–
932
(
1971
).
Google Scholar
 
34.
V. E.
 
Zakharov
 and
A. A.
 
Gelash
, “
Soliton on unstable condensate
,” preprint arXiv:1109.0620 [nlin.si] (
2011
).
Google Scholar
 
35.
V. E.
 
Zakharov
 and
A. A.
 
Gelash
, “
Nonlinear stage of modulational instability
,”
Phys. Rev. Lett.
 
111
,
054101
(
2013
).
https://doi.org/10.1103/PhysRevLett.111.054101
Google Scholar
Crossref
Search ADS
PubMed
 
36.
V. E.
 
Zakharov
 and
L. A.
 
Ostrovsky
, “
Modulation instability: The beginning
,”
Phys. D
 
238
,
540
–
548
(
2009
).
https://doi.org/10.1016/j.physd.2008.12.002
Google Scholar
Crossref
Search ADS
 
37.
V. E.
 
Zakharov
 and
A. B.
 
Shabat
, “
Exact theory of two-dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media
,”
Sov. Phys. JETP
 
34
,
62
–
69
(
1972
).
Google Scholar
 
38.
A. K.
 
Zvezdin
 and
A. F.
 
Popkov
, “
Contribution to the nonlinear theory of magnetostatic spin waves
,”
Sov. Phys. JETP
 
57
,
350
–
355
(
1983
).
Google Scholar
 
© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
You do not currently have access to this content.