Miura Transformations between Sokolov-Shabat's Equation and the Discrete MKdV Equation

Narita, Kazuaki

doi:10.1143/jpsj.66.4047

Cited by 5 publications

(8 citation statements)

References 4 publications

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“…Because fractional‐order models may well describe the dependent process of function development, 26 we consider the space–time fractional discrete coupled nonlinear Schrödinger equations (STFCDNLSEs) as

\begin{matrix} lefttrue \\ {italiciD}_{t}^{α} u_{n} + (u_{n - 1} + u_{n + 1}) [1 + β ({(||, u_{n})}^{2} + {(||, v_{n})}^{2})] - 2 u_{n} = 0, \\ {italiciD}_{t}^{α} v_{n} + (v_{n - 1} + v_{n + 1}) [1 + β ({(||, u_{n})}^{2} + {(||, v_{n})}^{2})] - 2 v_{n} = 0, \end{matrix}

where u n and v n denote complex variables defined as the site index n with integer values and β = ± 1. If α = 1, we obtain the discrete coupled NLSEs in Dai et al 3 Further, if β = 1 in Equation , we get the discrete coupled NLSEs in Narita 25 . When u n = 0( orv n = 0), Equation becomes the fractional Ablowitz–Ladik (AL) system 21 .…”

Section: Model and Mathematical Preparationsmentioning

confidence: 99%

Discrete soliton solutions of the fractional discrete coupled nonlinear Schrödinger equations: Three analytical approaches

Wang

Dai

2021

Math Methods in App Sciences

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The fractional discrete coupled nonlinear Schrödinger equations are solved on account of the modified Riemann–Liouville fractional derivative and Mittag–Leffler function. By using the fractional generalized Riccati method, generalized Mittag–Leffler function method and fractional generalized tanh–sech function method, some new analytical discrete solutions constructed by generalized trigonometric and hyperbolic functions are obtained, including discrete fractional bright soliton, dark soliton, combined soliton, and periodic solutions. In order to illustrate the effect of fractional order parameter on dynamics of fractional discrete soliton, some results in this study are illustrated graphically. Results indicate that although the combined wave is made up of singular coth function, it does not exhibit the singular property in the discrete case. Moreover, two kinds of scalar soliton solutions are found. These results could be of great significance to further study complex nonlinear discrete physical phenomena.

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\begin{matrix} lefttrue \\ {italiciD}_{t}^{α} u_{n} + (u_{n - 1} + u_{n + 1}) [1 + β ({(||, u_{n})}^{2} + {(||, v_{n})}^{2})] - 2 u_{n} = 0, \\ {italiciD}_{t}^{α} v_{n} + (v_{n - 1} + v_{n + 1}) [1 + β ({(||, u_{n})}^{2} + {(||, v_{n})}^{2})] - 2 v_{n} = 0, \end{matrix}

Section: Model and Mathematical Preparationsmentioning

confidence: 99%

Discrete soliton solutions of the fractional discrete coupled nonlinear Schrödinger equations: Three analytical approaches

Wang

Dai

2021

Math Methods in App Sciences

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“…Narita found two kinds of Miura transformations q n = (u n−1 + u n+1 )u n − 2u n−1 u n+1 (u n−1 − u n+1 )u n , q n = u n−1 − 2u n + u n+1 u n−1 − u n+1 between the equation (1) and Sokolov-Shabat equation [10] u n = 4(u n−1 − u n )(u n − u n+1 )/(u n−1 − u n+1 ).…”

Section: Introductionmentioning

confidence: 99%

“…between the equation (1) and Sokolov-Shabat equation [10] un = 4(u n−1 − u n )(u n − u n+1 )/(u n−1 − u n+1 ).…”

Section: Introductionmentioning

confidence: 99%

Long-Time Asymptotic Behavior for the Discrete Defocusing mKdV Equation

Chen¹,

Fan²

2019

J Nonlinear Sci

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“…It is known that self-dual network can also be reduced to the discrete analogue of the mKdV equation [7]. we note that there are many other differential-difference equations which can be transformed into the dmKdV equation [8][9][10][11][12][13][14][15]. The dmKdV equation has widely applications in the fields as plasma physics, electromagnetic waves in ferromagnetic, antiferromagnetic or dielectric systems, and can be solved by the method of inverse scattering transform, Hirota bilinear, Algebro-geometric approach and others [3,6,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

The discrete mKdV equation revisited: a Riemann-Hilbert approach

Zhu,

Geng,

Kuang

2013

Preprint

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We study the plus and minus type discrete mKdV equation. Some different symmetry conditions associated with two Lax pairs are introduced to derive the matrix Riemann-Hilbert problem with zero. By virtue of regularization of the Riemann-Hilbert problem, we obtain the complex and real solution to the plus type discrete mKdV equation respectively. Under the gauge transformation between the plus and minus type, the solutions of minus type can be obtained in terms of the given plus ones.

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Miura Transformations between Sokolov-Shabat's Equation and the Discrete MKdV Equation

Cited by 5 publications

References 4 publications

Discrete soliton solutions of the fractional discrete coupled nonlinear Schrödinger equations: Three analytical approaches

Discrete soliton solutions of the fractional discrete coupled nonlinear Schrödinger equations: Three analytical approaches

Long-Time Asymptotic Behavior for the Discrete Defocusing mKdV Equation

The discrete mKdV equation revisited: a Riemann-Hilbert approach

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