Shock Wave and Hole Type Soliton of Nonlinear Self-Dual Network Equation

Noguchi, Akira; Watanabe, Hideaki; Sakai, Kiyoshi

doi:10.1143/jpsj.43.1441

Cited by 10 publications

(3 citation statements)

References 10 publications

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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].…”

Section: Introductionmentioning

confidence: 98%

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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Section: Introductionmentioning

confidence: 98%

The discrete mKdV equation revisited: a Riemann-Hilbert approach

Zhu,

Geng,

Kuang

2013

Preprint

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“…The infinitely many conservation laws of equation (1.1) with σ = 1 were given [55,56]. The IST was used to find some shock wave and hole-type soliton solutions with non-zero asymptotic values of equation (1.1) with σ = −1 [57]. Equation (1.1) with σ = 1 and saturable nonlinearity was also discussed by the IST to find four types of solitons with non-zero asymptotic values [58].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours

2020

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The nonlinear self-dual network equations that describe the propagations of electrical signals in nonlinear LC self-dual circuits are explored. We firstly analyse the modulation instability of the constant amplitude waves. Secondly, a novel generalized perturbation ( M , N − M )-fold Darboux transform (DT) is proposed for the lattice system by means of the Taylor expansion and a parameter limit procedure. Thirdly, the obtained perturbation (1, N − 1)-fold DT is used to find its new higher-order rational solitons (RSs) in terms of determinants. These higher-order RSs differ from those known results in terms of hyperbolic functions. The abundant wave structures of the first-, second-, third- and fourth-order RSs are exhibited in detail. Their dynamical behaviours and stabilities are numerically simulated. These results may be useful for understanding the wave propagations of electrical signals.

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“…(4) has been explored extensively in works [14,15]. As far as we could verify, relatively less work is being performed for the symbolic computation of exact solutions to fractional-type DDEs while there has been a considerable amount of work done in finding exact solutions to polynomial DDEs.…”

Section: Introductionmentioning

confidence: 99%

Construction of exact solutions for fractional-type difference-differential equations via symbolic computation

Aslan

2013

Computers & Fluids

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a b s t r a c tThis paper deals with fractional-type difference-differential equations by means of the extended simplest equation method. First, an equation related to the discrete KdV equation is considered. Second, a system related to the well-known self-dual network equations through a real discrete Miura transformation is analyzed. As a consequence, three types of exact solutions (with the aid of symbolic computation) emerged; hyperbolic, trigonometric and rational which have not been reported before. Our results could be used as a starting point for numerical procedures as well.

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Shock Wave and Hole Type Soliton of Nonlinear Self-Dual Network Equation

Cited by 10 publications

References 10 publications

The discrete mKdV equation revisited: a Riemann-Hilbert approach

The discrete mKdV equation revisited: a Riemann-Hilbert approach

Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours

Construction of exact solutions for fractional-type difference-differential equations via symbolic computation

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