We develop a revised Riemann–Hilbert problem (RHP) to the Fokas–Lenells (FL) equation with a zero boundary condition, satisfying the normalization condition, and the potential of the FL equation is recovered from the asymptotic behavior of RHP when the spectral parameter goes to zero. Under the reflection-less situation, we consider the RHP with 2N simple poles and two Nth order poles, respectively, and obtain the explicit formulas of Nth order soliton and positon solutions. As applications, the first-order soliton, the second-order soliton, and positon are displayed. Additionally, the collisions of N solitons are studied, and the phase shift and space shift are displayed.
REFERENCES
1.
A. S.
Fokas
, “On a class of physically important integrable equations
,” Phys. D
87
, 145
–150
(1995
).2.
J.
Lenells
and A. S.
Fokas
, “On a novel integrable generalization of the nonlinear Schrödinger equation
,” Nonlinearity
22
, 11
–27
(2009
).3.
J.
Lenells
and A. S.
Fokas
, “An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons
,” Inverse Probl.
25
, 115006
(2009
).4.
J.
Lenells
, “Exactly solvable model for nonlinear pulse propagation in optical fibers
,” Stud. Appl. Math.
123
, 215
–232
(2009
).5.
J.
Lenells
, “Dressing for a novel integrable generalization of the nonlinear Schrödinger equation
,” J. Nonlinear Sci.
20
, 709
–722
(2010
).6.
R.
Camassa
and D. D.
Holm
, “An integrable shallow water equation with peaked solitons
,” Phys. Rev. Lett.
71
, 1661
–1664
(1993
).7.
L.
Ai
and J.
Xu
, “On a Riemann–Hilbert problem for the Fokas–Lenells equation
,” Appl. Math. Lett.
87
, 57
–63
(2019
).8.
Y.
Zhao
and E.
Fan
, “Inverse scattering transformation for the Fokas–Lenells equation with nonzero boundary conditions
,” J. Nonlinear Math. Phys.
28
, 38
–52
(2020
).9.
J.
He
, S.
Xu
, and K.
Porsezian
, “Rogue waves of the Fokas–Lenells equation
,” J. Phys. Soc. Jpn.
81
, 124007
(2012
).10.
S.
Xu
, J.
He
, Y.
Cheng
, and K.
Porseizan
, “The n-order rogue waves of Fokas–Lenells equation
,” Math. Methods Appl. Sci.
38
, 1106
–1126
(2015
).11.
P.
Zhao
, E.
Fan
, and Y.
Hou
, “Algebro-geometric solutions and their reductions for the Fokas-Lenells hierarchy
,” J. Nonlinear Math. Phys.
20
, 355
–393
(2013
).12.
D. J.
Kaup
and A. C.
Newell
, “An exact solution for a derivative nonlinear Schrödinger equation
,” J. Math. Phys.
19
, 798
–801
(1978
).13.
G.-Q.
Zhou
and N.-N.
Huang
, “An N-soliton solution to the DNLS equation based on revised inverse scattering transform
,” J. Phys. A: Math. Theor.
40
, 13607
–13623
(2007
).14.
G.
Zhang
and Z.
Yan
, “The derivative nonlinear Schrödinger equation with zero/nonzero boundary conditions: Inverse scattering transforms and N-double-pole solutions
,” J. Nonlinear Sci.
30
, 3089
–3127
(2020
).15.
Y.
Zhang
, J.
Rao
, Y.
Cheng
, and J.
He
, “Riemann–hilbert method for the Wadati–Konno–Ichikawa equation: N simple poles and one higher-order pole
,” Phys. D
399
, 173
–185
(2019
).16.
Y.
Zhang
, X.
Tao
, and S.
Xu
, “The bound-state soliton solutions of the complex modified KdV equation
,” Inverse Probl.
36
, 065003
(2020
).17.
Z.
Zhang
and E.
Fan
, “Inverse scattering transform and multiple high-order pole solutions for the Gerdjikov–Ivanov equation under the zero/nonzero background
,” Z. Angew. Math. Phys.
72
, 153
(2021
).18.
X.
Ma
, “Riemann-Hilbert approach for a higher-order Chen–Lee–Liu equation with high-order poles
,” Commun. Nonlinear Sci. Numer. Simul.
114
, 106606
(2022
).19.
Y.
Zhang
, H.
Wu
, and D.
Qiu
, “Revised Riemann-Hilbert problem for the derivative nonlinear Schrödinger equation: Vanished boundary condition
,” Submitted.20.
Y.
Zhang
, N.
Wang
, D.
Qiu
, and J.
He
, “Explicit solitons of Kundu equation derived by Riemann-Hilbert problem
,” Phys. Lett. A
452
, 128476
(2022
).21.
D.
Qiu
and Y.
Zhang
, “The explicit bound-state soliton of Kundu equation derived by Riemann–Hilbert problem
,” Appl. Math. Lett.
135
, 108443
(2023
).22.
V. B.
Matveev
, “Generalized Wronskian formula for solutions of the KdV equations: First applications
,” Phys. Lett. A
166
, 205
–208
(1992
).23.
V. B.
Matveev
, “Positon-positon and soliton-positon collisions: KdV case
,” Phys. Lett. A
166
, 209
–212
(1992
).24.
S.
Novikov
, S. V.
Manakov
, L. P.
Pitaevskii
, and V. E.
Zakharov
, Theory of Solitons: The Inverse Scattering Method
(Springer Science and Business Media
, 1984
).25.
M. J.
Ablowitz
and P. A.
Clarkson
, Solitons, Nonlinear Evolution Equations and Inverse Scattering
(Cambridge University Press
, Cambridge, UK
, 1991
).26.
E.-G.
Fan
, Integrable System, Orthogonal Polynomial and Random Matrix: Riemann–Hilbert Approach
(The Science Press
, Beijing, China
, 2022
).27.
D.
Bilman
and R.
Buckingham
, “Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation
,” J. Nonlinear Sci.
29
, 2185
–2229
(2019
).28.
D.
Bilman
, L.
Ling
, and P. D.
Miller
, “Extreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchy
,” Duke Math. J.
169
, 671
–760
(2020
).29.
X.
Zhang
and L.
Ling
, “Asymptotic analysis of high-order solitons for the Hirota equation
,” Phys. D
426
, 132982
(2021
).30.
G.
Zhang
, L.
Ling
, and Z.
Yan
, “Multi-component nonlinear Schrödinger equations with nonzero boundary conditions: Higher-order vector Peregrine solitons and asymptotic estimates
,” J. Nonlinear Sci.
31
, 81
(2021
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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