Solutions for the Mikhailov-Shabat-Yamilov Difference-Differential Equations and Generalized Solutions for the Volterra and the Toda Lattice Equations

Narita, Kazuaki

doi:10.1143/ptp.99.337

Cited by 9 publications

(7 citation statements)

References 16 publications

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“…Taking into account eqs. (15)(16)(17) and using the Cauchy-Green formula we can write the following integral equations for the two Jost solutions…”

Section: Inverse Problemmentioning

confidence: 99%

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Continuous symmetries of the lattice potential KdV equation

Levi¹,

Petrera²

2007

J. Phys. A: Math. Theor.

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In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg-de Vries (lpKdV) equation. Using its associated spectral problem we construct the soliton solutions and the Lax technique enables us to provide infinite sequences of generalized symmetries. Finally, using a discrete symmetry of the lpKdV equation, we construct a large class of non-autonomous generalized symmetries.

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“…Taking into account eqs. (15)(16)(17) and using the Cauchy-Green formula we can write the following integral equations for the two Jost solutions…”

Section: Inverse Problemmentioning

confidence: 99%

“…, where Ω m (κ) is a normalization function, and taking into account the asymptotic behaviours (15)(16)(17) of the Jost functions, we find from eq. ( 21) the following time evolution of…”

Section: Inverse Problemmentioning

confidence: 99%

Continuous symmetries of the lattice potential KdV equation

Levi¹,

Petrera²

2007

J. Phys. A: Math. Theor.

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“…In Ref. [48], some new generalized soliton solutions for the Toda lattice equations based on the invariance of Gibbon and Tabor's equation under the fractional linear transformation have been given. In Ref.…”

Section: Introductionmentioning

confidence: 99%

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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“…As far as we could verify, DDEs of such types received relatively little attention. To achieve our goal, we consider the following time‐fractional DDEs

\frac{d^{α} u_{n}}{d t^{α}} = \frac{u_{n - 1} - u_{n + 1}}{1 - u_{n - 1} + u_{n + 1}}, 0 < α \leq 1,

\frac{d^{α} u_{n}}{d t^{α}} = \frac{4 ((), u_{n - 1} - u_{n + 1}) u_{n}^{2} ((), 1 - u_{n}^{2})}{((), u_{n - 1} + u_{n}) ((), u_{n} + u_{n + 1})}, 0 < α \leq 1,

which are proposed in . Equation and Equation are related to the Volterra equation and discrete modified KdV equation through dependent variable transformations, respectively.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

Aslan

2016

Math Methods in App Sciences

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Dynamical behavior of many nonlinear systems can be described by fractional-order equations. This study is devoted to fractional differential-difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)-expansion method coupled with the so-called fractional complex transform. The solution procedure is elucidated through two generalized time-fractional differential-difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational.

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Solutions for the Mikhailov-Shabat-Yamilov Difference-Differential Equations and Generalized Solutions for the Volterra and the Toda Lattice Equations

Cited by 9 publications

References 16 publications

Continuous symmetries of the lattice potential KdV equation

Continuous symmetries of the lattice potential KdV equation

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

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

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