Elastic Interaction and Conservation Laws for the Nonlinear Self-Dual Network Equation in Electric Circuit

Wen, Xiao-Yong

doi:10.1143/jpsj.81.114006

Cited by 31 publications

(25 citation statements)

References 29 publications

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“…The present paper addresses this issue for discrete coupled equations (1). The key feature of such a technique is the fact that the Lax pair associated with the integrable nonlinear differential-difference equations is covariant under the gauge transformation [36,38].…”

Section: Introductionmentioning

confidence: 99%

Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability

Wen

Yan

Malomed

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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An integrable system of two-component nonlinear Ablowitz-Ladik equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, the perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.

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Section: Introductionmentioning

confidence: 99%

Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability

Wen

Yan

Malomed

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Other types of solitary wave and periodic solutions of equation (1.1) were obtained [61][62][63][64]. Recently, we constructed the N-soliton solutions for equation (1.1) with σ = 1 via the N-fold DT [65].…”

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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“…Exact solutions can make people understand deeply the physical mechanism of the natural phenomena described by NLEs. Some methods for constructing explicit exact solutions of NLEs, such as the inverse scattering method [1-4, 18, 19], Hirota method [13,20], Bäcklund transformation [21][22][23], and Darboux transformation (DT) [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], have been developed. Among them, the DT based on the Lax pair is an algebraic method which is an effective tool to solve the Lax integrable NLEs.…”

Section: Introductionmentioning

confidence: 99%

“…[31][32][33]), and the DT with the N -order Darboux matrix whose elements are the polynomials of the spectral parameters (see, e.g., Refs. [34][35][36][37][38][39][40][41]), etc. Theoretically, the solution obtained by a single DT (i.e., 1-fold DT which is the N -fold DT when N = 1) can be taken as the new starting point from which to derive another solution by making a DT again, hence a series of explicit solutions can be generated by iteration step by step [35][36][37].…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of bright and dark multi-soliton solutions for two higher-order Toda lattice equations for nonlinear waves

Liu

Wen

2018

Adv Differ Equ

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Discrete N-fold Darboux transformation (DT) is used to derive new bright and dark multi-soliton solutions of two higher-order Toda lattice equations. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied via numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such bright and dark multi-solitons without a noise. To compare the numerical evolution results with the classical Toda lattice equation, we also investigate the dynamical behaviors of the multi-soliton solutions for Toda lattice equation via numerical simulations, and we find that the multi-soliton solutions of Toda lattice equation have better stability and are more robust against a big noise than its two corresponding higher-order equations. The same small noise has different effect on the evolutions of the multi-soliton solutions for three different equations in the same hierarchy. The possible reason is that the higher-order nonlinear terms of the higher-order equation cause the instability of the wave propagation. The discrete generalized (n, N-n)-fold DTs are constructed and used to derive some discrete rational solutions of three equations, and a few mathematical features for such rational solutions are also discussed. Results might be helpful for understanding the propagation of nonlinear waves in soliton theory.

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Elastic Interaction and Conservation Laws for the Nonlinear Self-Dual Network Equation in Electric Circuit

Cited by 31 publications

References 29 publications

Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability

Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability

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

Dynamics of bright and dark multi-soliton solutions for two higher-order Toda lattice equations for nonlinear waves

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