Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation

Liu, Wei; Zheng, Xiaoxiao; Li, Xiliang

doi:10.1007/s11071-018-4482-9

Cited by 20 publications

(9 citation statements)

References 82 publications

(83 reference statements)

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“…We stress that the two solitons form a parallel pair, and their dynamical behavior is different from that of the cross solitons in nonlocal systems considered in Refs. 49, 50, 92. As can be seen in Figures 7(D)–(F), the two solitons pass through each other without changes in their velocity and waveforms, which suggests that there the interaction between the two solitons is strictly elastic, as in other integrable systems.…”

Section: General Line Solitons On Top Of a Constant Backgroundmentioning

confidence: 70%

“…Recently, general soliton solutions for a nonlocal NLS equation were produced using a combination of the Hirota's bilinear method and KP hierarchy reduction 91 . This finding has triggered rapid progress in studies of solitons in nonlocal systems 49,50,92,93 . Inspired by that work, we consider tau‐functions of the nonlocal Maccari system.Lemma Referring to the Sato theory 74,94–97 , the bilinear equations in the KP hierarchy

\begin{matrix} true \\ left (D_{x_{1}}^{2} - D_{x_{2}}) τ_{n + 1} \cdot τ_{n} = 0, \\ left (D_{x_{- 1}}^{2} + D_{x_{- 2}}) τ_{n + 1} \cdot τ_{n} = 0, \\ left (D_{x_{1}} D_{x_{- 1}} - 2) τ_{n} \cdot τ_{n} = - 2 τ_{n + 1} \cdot τ_{n - 1}, \end{matrix}

give rise to the following tau‐functions,

τ_{n} = \underset{1 \leq j, k \leq N}{trueprefixdet} false(m_{j, k}^{false(n false)} false),

with matrix elements

m_{i, j}^{false(n false)}

satisfying the following difference relations,

\begin{matrix} true \\ left \partial_{x_{1}} m_{j, k}^{false(n false)} = φ_{j}^{false(n false)} ψ_{k}^{false(n false)}, m_{j, k}^{false(n + 1 false)} = m_{j, k}^{false(n false)} + φ \end{matrix}

…”

Section: General Line Solitons On Top Of a Constant Backgroundmentioning

confidence: 99%

Section: General Line Solitons On Top Of a Constant Backgroundmentioning

confidence: 99%

“…Since then, the discrete versions [36][37][38][39] of Equation (1) and its various solutions [40][41][42][43][44][45] have been studied in depth. Subsequently, several extended one-dimensional and two-dimensional nonlocal nonlinear evolution equations have been introduced [46][47][48][49][50][51][52] . An issue of obvious interest is to generate a multidimensional integrable model with  -symmetry and explore interaction dynamics in it, which is a motivation for the present work.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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.

show abstract

Section: General Line Solitons On Top Of a Constant Backgroundmentioning

confidence: 70%

\begin{matrix} true \\ left (D_{x_{1}}^{2} - D_{x_{2}}) τ_{n + 1} \cdot τ_{n} = 0, \\ left (D_{x_{- 1}}^{2} + D_{x_{- 2}}) τ_{n + 1} \cdot τ_{n} = 0, \\ left (D_{x_{1}} D_{x_{- 1}} - 2) τ_{n} \cdot τ_{n} = - 2 τ_{n + 1} \cdot τ_{n - 1}, \end{matrix}

give rise to the following tau‐functions,

τ_{n} = \underset{1 \leq j, k \leq N}{trueprefixdet} false(m_{j, k}^{false(n false)} false),

with matrix elements

m_{i, j}^{false(n false)}

satisfying the following difference relations,

\begin{matrix} true \\ left \partial_{x_{1}} m_{j, k}^{false(n false)} = φ_{j}^{false(n false)} ψ_{k}^{false(n false)}, m_{j, k}^{false(n + 1 false)} = m_{j, k}^{false(n false)} + φ \end{matrix}

…”

Section: General Line Solitons On Top Of a Constant Backgroundmentioning

confidence: 99%

Section: General Line Solitons On Top Of a Constant Backgroundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Cao¹,

Cheng²,

Malomed

et al. 2021

Stud Appl Math

View full text Add to dashboard Cite

show abstract

“…We stress that the two solitons form a parallel pair, and their dynamical behavior is different from that of the cross solitons in nonlocal systems considered in Refs. [49,50,92]. As can be seen in Fig.…”

Section: General Line Solitons On Top Of a Constant Backgroundmentioning

confidence: 73%

Rogue waves and lumps on the non-zero background in the PT -symmetric nonlocal Maccari system

Cao,

Cheng,

Malomed

et al. 2020

Preprint

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In this paper, the PT -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 solutions and semi-rational ones, composed of hyperbolic line rogue wave and periodic line waves are also derived in the long-wave limit. The semi-rational 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 semi-rational 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 semi-rational 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.

show abstract

Interaction of wave structure in the $$\mathcal{P}\mathcal{T}$$-symmetric $$(3\,+\,1)$$-dimensional nonlocal Mel’nikov equation and their applications

Cao

Tian

Wazwaz

et al. 2023

Z. Angew. Math. Phys.

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Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation

Cited by 20 publications

References 82 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

Rogue waves and lumps on the non-zero background in the PT -symmetric nonlocal Maccari system

Interaction of wave structure in the $$\mathcal{P}\mathcal{T}$$-symmetric $$(3\,+\,1)$$-dimensional nonlocal Mel’nikov equation and their applications

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