Two-phase solutions of focusing NLS equation are classically constructed out of an appropriate Riemann surface of genus two and expressed in terms of the corresponding theta-function. We show here that in a certain limiting regime, such solutions reduce to some elementary ones called “Solitons on unstable condensate.” This degeneration turns out to be conveniently studied by means of basic tools from the theory of Riemann-Hilbert problems. In particular, no acquaintance with Riemann surfaces and theta-function is required for such analysis.
REFERENCES
1.
Belokolos
, E. D.
, Bobenko
, A. I.
, Enol’ski
, V. Z.
, Its
, A. R.
, and Matveev
, V. B.
, Algebro-Geometric Approach to Nonlinear Integrable Equations
, Springer Series in Nonlinear Dynamics
(Springer-Verlag
, 1994
).2.
Biondini
, G.
and Kovačič
, G.
, “Inverse scattering transform for the focusing nonlinear Schödinger equation with nonzero boundary conditions
,” J. Math. Phys.
55
, 031506
(2014
).3.
Computational Approach to Riemann Surfaces, Lecture Notes in Mathematics Vol. 2013, edited by A. I. Bobenko and C. Klein (Springer-Verlag, 2011).
4.
Bujalance
, E.
, Cirre
, F. J.
, Gamboa
, J. M.
, and Gromadzki
, G.
, in Symmetries of Compact Riemann Surfaces
(Springer Science & Business Media
, 2010
), Vol. 2007
.5.
Calini
, A.
and Ivey
, T.
, “Finite-gap solutions of the vortex filament equation: Genus one solutions and symmetric solutions
,” J. Nonlinear Sci.
15
, 321
-361
(2005
).6.
Chabchoub
, A.
, Vitanov
, N.
, and Hoffmann
, N.
, “Experimental evidence for breather type dynamics in freak waves
,” Proc. Appl. Math. Mech.
10
, 495
-496
(2010
).7.
Chiao
,R. Y.
, Garmire
,E.
, and Townes
,C. H.
, “Self-trapping of optical beams
,” Phys. Rev. Lett.
13
, 479
(1964
);Erratum
Chiao
, R. Y.
, Garmire
, E.
, and Townes
, C. H.
, Phys. Rev. Lett.
14
, 1056
(1965
).8.
Deift
, P.
, in Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
(American Mathematical Society
, 2000
), Vol. 3
.9.
Dubrovin
, B. A.
, “Theta functions and non-linear equations
,” Russ. Math. Surv.
36
(2
), 11
-92
(1981
).10.
Dubrovin
, B.
, “On analytic families of invariant tori for PDEs
,” Astérisque
297
, 35
-65
(2004
).11.
Dysthe
, B. K.
and Trulsen
, K.
, “Note on breather type solutions of the NLS as models for freak-waves
,” Phys. Scr.
T82
, 48
-52
(1999
).12.
Farkas
, H. M.
and Kra
, I.
, Riemann Surfaces
(Springer
, New York
, 1992
).13.
Fay
, J.
, Theta Functions on Riemann Surfaces
, Lecture Notes in Mathematics
Vol. 352
(Springer-Verlag
, 1973
).14.
Fokas
, A. S.
, Its
, A. R.
, Kapaev
, A. A.
, and Novokshenov
, V. Yu.
, “Painlevé transcendents. The Riemann–Hilbert approach
,” in Mathematical Surveys and Monographs
(AMS
, 2006
), Vol. 128
.15.
Giavedoni
, P.
, “Period matrices of real Riemann surfaces and fundamental domains
,” SIGMA
9
, 062
(2013
).16.
Giavedoni
, P.
, “Topics on the two-phase solutions of the focusing nonlinear Schrödinger equation
,” Ph.D. thesis (2012
).17.
Its
, A. R.
and Kotlyarov
, V. P.
, “Explicit expressions for the solutions of nonlinear Schrödinger equation
,” Dockl. Akad. Nauk. SSSR, S. A
965
, 11
(1976
).18.
Jimbo
, M.
, Miwa
, T.
, and Ueno
, K.
, “Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (I)
,” Physica D
2
, 306
-352
(1981
).19.
Korotkin
, D.
, “Solution of an arbitrary matrix Riemann-Hilbert problem with quasi-permutation monodromy matrices
,” Math. Ann.
329
(2
), 335
-364
(2004
).20.
Ohta
, Y.
and Yang
, J.
, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation
,” Proc. R. Soc. A
468
, 1716
-1740
(2012
).21.
Rogers
, C.
and Schief
, W. K.
, “Bcklund and Darboux transformations. Geometry and modern applications in soliton theory
,” in Cambridge Texts in Applied Mathematics
(Cambridge University Press
, Cambridge
, 2002
).22.
23.
Zakharov
, V. E.
and Shabat
, A. B.
, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media
,” Sov. Phys. JETP
34
, 62
-69
(1972
).24.
Zakharov
, V. E.
and Gelash
, A. A.
, “Soliton on unstable condensate
,” e-print arXiv:1109.0620 (2011).25.
Zakharov
, V. E.
, “Stability of periodic waves of finite amplitude on the surface of a deep fluid
,” J. Appl. Mech. Tech. Phys.
9
, 190
-194
(1968
).© 2015 AIP Publishing LLC.
2015
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