Dynamical Behavior of Solution in Integrable Nonlocal Lakshmanan—Porsezian—Daniel Equation

Liu, Wei; Qiu, Deqin; Wu, Zhiwen; He, Jingsong

doi:10.1088/0253-6102/65/6/671

Cited by 69 publications

(25 citation statements)

References 30 publications

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“…THEOREM 2. The nonlocal DSII equation possesses two N th-order rational solutions A and Q given by (34) with the two following polynomials f and g:…”

Section: Rational Solutions Of the Nonlocal Dsii Equationmentioning

confidence: 99%

“…Note that two types of rogue wave solutions for the nonlocal DSI equation were obtained formally in [22], and Darboux transformations and global explicit solutions for the x-nonlocal DSI equation were given in [23]. Besides, a lot of work was done after that for these equations and other nonlocal equations [19,20,[23][24][25][26][27][28][29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

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Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Rao¹,

Cheng

2017

Stud Appl Math

183

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In this paper, the partially party-time (PT ) symmetric nonlocal Davey-Stewartson (DS) equations with respect to x is called x-nonlocal DS equations, while a fully PT symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x-nonlocal DS equations, the usual (2 + 1)-dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)-dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.

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“…THEOREM 2. The nonlocal DSII equation possesses two N th-order rational solutions A and Q given by (34) with the two following polynomials f and g:…”

Section: Rational Solutions Of the Nonlocal Dsii Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Rao¹,

Cheng

2017

Stud Appl Math

183

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“…Since the LDP model plays a significant role in portraying several complex phenomena, many authors find and analysed the solution using analytical as well as numerical schemes. For instance, Arshed et al 54 find the optical solitons for the considered model with the help of exp-(ϕ(ξ))-expansion method in birefringent fibres; the bright optical solitons have been illustrated by Alqahtani et al 55 for LPD model using semi-inverse variational principle, the group invariant solutions, and optical solitons in optical fibres; and PCF is presented by Bansal et al, 56 the optical solitons in birefringent fibres for the considered model achieved by Ekici 57 using the extended Jacobi's elliptic function expansion method; Guzman et al 58 applied the method of undetermined coefficients in order to find the optical solitons with polarization mode dispersion for the considered complex non-linear problem; the singular and dark optical solitons are obtained by Javid and Raza 59 ; singular and dark optical solitons have been obtained by Jawad et al 60 ; and many others [61][62][63][64] presented some interesting results. However, these techniques have their one limitation like huge computation, perturbation, and discretization.…”

Section: Introductionmentioning

confidence: 99%

An efficient analytical approach for fractional Lakshmanan‐Porsezian‐Daniel model

et al. 2020

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In this paper, the q‐homotopy analysis transform method (q‐HATM) is applied to find the solution for the fractional Lakshmanan‐Porsezian‐Daniel (LPD) model. The LPD model is the generalization of the non‐linear Schrödinger (NLS) equation. The proposed method is graceful fusions of Laplace transform technique with q‐homotopy analysis scheme, and the derivative is considered in Caputo sense. In order to validate and illustrate the efficiency of the proposed method, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured for the three different cases in terms of 3D and contour plots for diverse values of the fractional order. The obtained results confirm that the future method is easy to implement, highly methodical, and very effective to analyse the behaviour of complex non‐linear fractional differential equations exist in the connected areas of science and engineering.

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“…[20,29,33]. The integrable nonlocal LPD equation whose the potential functions satisfy r(x, t) = q * (−x, t) is studied via Darboux transformation in [21]. The authors demonstrated the integrability of the nonlocal LPD equation, provided its Lax pair, and presented its rational soliton solutions and self-potential function.…”

mentioning

confidence: 99%

Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

Xun¹,

Tian²,

Zhang³

2022

DCDS-B

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<p style='text-indent:20px;'>In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. We successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering problem is analyzed, and the potential function is recovered. By introducing the special conditions of reflectionless case, the time-periodic soliton solutions formula of the equation is derived successfully. Take <inline-formula><tex-math id="M1">\begin{document}$ J = \overline{J} = 1,2,3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.</p>

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Dynamical Behavior of Solution in Integrable Nonlocal Lakshmanan—Porsezian—Daniel Equation

Cited by 69 publications

References 30 publications

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

An efficient analytical approach for fractional Lakshmanan‐Porsezian‐Daniel model

Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

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