Novel higher-order rational solitons and dynamics of the defocusing integrable nonlocal nonlinear Schrödinger equation via the determinants

Zhang, Guoqiang; Yan, Zhenya; Chen, Yong

doi:10.1016/j.aml.2017.02.002

Cited by 45 publications

(26 citation statements)

References 19 publications

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“…With the increase of the interaction strength, the central region of the cross‐shaped RW creates two sharp peaks at

t \approx - 3 / 4

. The cross‐shaped RW is separated into two hyperbolic line RWs at

t \approx 0

, whose behavior is similar to that of the rational‐soliton in the

(1 + 1)

‐dimensional system 88,89 ; then they merge back into the constant background, the strong interaction resulting in the change of the waveform. We stress that the dynamical behavior of the higher order line RWs of nonlocal Maccari system () is different from that of the corresponding higher order line RWs in other nonlocal systems, such as the nonlocal two‐dimensional NLS equation 48,90 and nonlocal DS equation 31,70 .…”

Section: General Line Rws On Top Of the Periodic Line‐wave Backgroundmentioning

confidence: 96%

Rogue waves and lumps on the nonzero background in the ‐symmetric nonlocal Maccari system

Cao¹,

Cheng²,

Malomed

et al. 2021

Stud Appl Math

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In this paper, the scriptPT‐symmetric version of the Maccari system is introduced, which can be regarded as a two‐dimensional generalization of the defocusing nonlocal nonlinear Schrödinger equation. Various exact solutions of the nonlocal Maccari system are obtained by means of the Hirota bilinear method, long‐wave limit, and Kadomtsev–Petviashvili (KP) hierarchy method. Bilinear forms of the nonlocal Maccari system are derived for the first time. Simultaneously, a new nonlocal Davey‐Stewartson–type equation is derived. Solutions for breathers and breathers on top of periodic line waves are obtained through the bilinear form of the nonlocal Maccari system. Hyperbolic line rogue wave (RW) solutions and semirational ones, composed of hyperbolic line RW and periodic line waves are also derived in the long‐wave limit. The semirational solutions exhibit a unique dynamical behavior. Additionally, general line soliton solutions on constant background are generated by restricting different tau‐functions of the KP hierarchy, combined with the Hirota bilinear method. These solutions exhibit elastic collisions, some of which have never been reported before in nonlocal systems. Additionally, the semirational solutions, namely, (i) fusion of line solitons and lumps into line solitons and (ii) fission of line solitons into lumps and line solitons, are put forward in terms of the KP hierarchy. These novel semirational solutions reduce to 2N‐lump solutions of the nonlocal Maccari system with appropriate parameters. Finally, different characteristics of exact solutions for the nonlocal Maccari system are summarized. These new results enrich the structure of waves in nonlocal nonlinear systems, and help to understand new physical phenomena.

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“…With the increase of the interaction strength, the central region of the cross‐shaped RW creates two sharp peaks at

t \approx - 3 / 4

. The cross‐shaped RW is separated into two hyperbolic line RWs at

t \approx 0

, whose behavior is similar to that of the rational‐soliton in the

(1 + 1)

Section: General Line Rws On Top Of the Periodic Line‐wave Backgroundmentioning

confidence: 96%

Rogue waves and lumps on the nonzero background in the ‐symmetric nonlocal Maccari system

Cao¹,

Cheng²,

Malomed

et al. 2021

Stud Appl Math

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“…In this section, we will establish the determinant representation of the N -fold DT for Eq.(2). The spectral problem (6) can transform to Ψ [2] x = U [2] Ψ [2] , Ψ…”

Section: N -Fold Dt Of Reverse-space-time Dnls Equationmentioning

confidence: 99%

“…Nonlinear evolution equations play an important role and their solutions have been a hot research spot, including soliton [1,2,3,4], breather [5,6,7], rogue wave [8,9,10,11] and others [12,13,14,15]. The derivative nonlinear Schrödinger equation (DNLS) [16,17,18,19] iq t − q xx + i(q 2 q * ) x = 0, (…”

Section: Introductionmentioning

confidence: 99%

Breathers and Rogue Waves On The Double-Periodic Background for The Reverse-Space-Time Derivative Nonlinear Schrödinger Equation

Zhou

Chen

2021

Preprint

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The dynamic behaviors of the solutions for the reverse-space-time derivative nonlinear Schrödinger equation are studied by Darboux transformation. The breathers on the periodic and double-periodic background are derived by the N -fold Darboux transformation. The rogue waves on the periodic and double-periodic background are constructed by the generalized Darboux transformation. It is worth mentioning that the breathers and rogue waves on double-periodic background based on the plane wave seed solution are first constructed. The two peak, four peak rogue waves on the double-periodic background are found. And the rogue waves on the double-periodic background can be transformed into the classical rogue wave on the plane wave background with a special reduction relation.

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“…In this subsection, we use the generalized DT [19,[40][41][42]48,49] to derive dark-bright mixed highorder soliton solutions for equation (1.3). Let the spectral parameter λ = λ 1 + , where is a small parameter and (−2 Im(λ 1 )/Im(μ…”

Section: (C) Dark-bright Mixed High-order Soliton Solutionsmentioning

confidence: 99%

“…A natural problem is that how to use the Darboux transformation (DT) to analyse the soliton and breather structures from the viewpoint of the spectral parameter in the Lax pair in detail. Recently, the DT has been extended to derive multi-bright-bright and multi-dark-dark solitons for the nCNLS equations [39], high-order rational solitons [40][41][42] of the non-local NLS equation and its two-component case [43,44], and the dark-dark multi-solitons of the defocusing AB system [45]. To the best of our knowledge, the dark-bright mixed solitons and breathers, as well as their semi-rational extensions for the nCNLS equations (1.3) were not studied by the DT before.…”

Section: Introductionmentioning

confidence: 99%

Then-component nonlinear Schrödinger equations: dark–bright mixedN- and high-order solitons and breathers, and dynamics

Zhang

Yan

2018

Proc. R. Soc. A.

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The general n -component nonlinear Schrödinger equations are systematically investigated with the aid of the Darboux transformation method and its extension. Firstly, we explore the condition of the existence for dark–bright mixed soliton solutions and derive an explicit formula of dark–bright mixed multi-soliton solutions in terms of the determinant. Secondly, we present the formula of dark–bright mixed high-order semi-rational solitons, and predict their general N th-order wave structures. Thirdly, we investigate the wing-shaped structures of breather. Finally, we perform the numerical simulations for some representative solitons to study their dynamical behaviours.

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Novel higher-order rational solitons and dynamics of the defocusing integrable nonlocal nonlinear Schrödinger equation via the determinants

Cited by 45 publications

References 19 publications

Rogue waves and lumps on the nonzero background in the ‐symmetric nonlocal Maccari system

Rogue waves and lumps on the nonzero background in the ‐symmetric nonlocal Maccari system

Breathers and Rogue Waves On The Double-Periodic Background for The Reverse-Space-Time Derivative Nonlinear Schrödinger Equation

Then-component nonlinear Schrödinger equations: dark–bright mixedN- and high-order solitons and breathers, and dynamics

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