On a generalized Ablowitz–Kaup–Newell–Segur hierarchy in inhomogeneities of media: soliton solutions and wave propagation influenced from coefficient functions and scattering data

Journal of Low Frequency Noise, Vibration and Active Control

2018

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Three models are investigated in this paper, including the generalized nonlinear Schr€ odinger equation with distributed coefficients, the time-dependent-coefficient nonlinear Whitham-Broer-Kaup system and the fractional nonlinear vibration governing equation of an embedded single-wall carbon nanotube. With an analytical method, the aim of this paper is to exactly solve these models. As a result, some explicit and exact solutions which include hyperbolic function solutions, trigonometric function solutions and rational solutions are obtained. To gain more insights into the obtained exact solutions, dynamical evolutions with nonlinear vibrations of the amplitudes are simulated by selecting oscillation functions, noises, coefficient functions and fractional orders. It is graphically shown in the dynamical evolutions that the nonlinear vibrations of the amplitudes are influenced not only by the coefficient functions but also by the oscillation functions, noises and fractional orders.

Section: Resultsmentioning

confidence: 83%

Analytical insights into three models: Exact solutions and nonlinear vibrations

Journal of Low Frequency Noise, Vibration and Active Control

2018

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“…To gain more insights into the non-isospectral parameter (2) and the obtained n-soliton solutions (21), we set 1 nn  and then consider their dynamical evolutions and spatial structures. In this case, eqs.…”

Section: Dynamical Evolutions and Spatial Structuresmentioning

confidence: 99%

“…Thanks to the powerful VIM, HPM, and EXP method proposed by Ji-Huan He, a large number of exact and approximate solutions have been obtained in different fields. Recently, Zhang and Hong [21] extended the IST method to the generalized isospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, the variable-coefficient non-isospectral --------------Toda lattice hierarchy [22], the generalized non-isospectral AKNS hierarchies [23][24][25][26][27], the mixed spectral KdV hierarchy with self-consistent sources [28] and the mixed spectral AKNS hierarchies [29][30][31][32][33], supersymmetric KdV equation [34], respectively.…”

Section: Introductionmentioning

confidence: 99%

Inverse scattering transform for new mixed spectral Ablowitz-Kaup-Newell-Segur equations

You

2020

Therm sci

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The inverse scattering transform plays a very important role in promoting the development of analytical methods to solve non-linear PDE exactly. In this paper, new and more general mixed spectral Ablowitz-Kaup-Newell-Segur equations are derived and solved by embedding a novel time-varying spectral parameter in-to an associated linear problem and the inverse scattering transform. As a result, new exact solutions and n-soliton solutions are obtained. To gain more insights into the embedded spectral parameter and the obtained solutions, some dynamical evolutions, and spatial structures are simulated. It is shown that the derived Ablowitz-Kaup-Newell-Segur equations are Lax integrable and the obtained soliton solutions possess time-varying amplitudes.

“…In the field of nonlinear mathematical physics, many effective methods have been presented for constructing soliton solutions, such as Darboux transformation [1], Hirota bilinear method [2][3][4][5][6], inverse scatting transform (IST) [7][8][9][10], and others [11][12][13]. The RHM [14], seen as a modern version of the IST, is systematic method for both solutions and integrability of soliton equations.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert method and soliton dynamics for a mixed spectral complex mKdV equation with time-varying coefficients

Zhou

2023

Nonlinear Dyn

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This paper reports the feasibility of using Riemann-Hilbert method (RHM) to reveal the soliton dynamics with variable velocity and amplitude in mixed spectral nonlinear integrable systems. To achieve this goal, we consider a novel mixed spectral complex mKdV equation (mscmKdV) with time-varying coefficients. Firstly, Lax representation of the timevarying coefficient mscmKdV equation is first abstracted from Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problem equipped with a mixed spectrum. With the help of the Lax representatoin, a Riemann-Hilbert problem (RHP) associating with the mscmKdV equation is then established. Due to the confirmation of solvability of the RHP, scattering data used for recostructing the potential are further determined, and N-soliton solution of the mscmKdV equation is finally obtained. In addition, the one-and two-soliton solutions are shown to explore the characteristics of soliton dynamics in the mscmKdV equation. It is revealed fromthe mscmKdV equation that compared with the constant velocity and amplitude propagation of a soliton in the background of constant-coefficient isospectral model, the one of variablecoefficient isospectral system propagates with variable velocity but constant amplitude, while for the case of non-spectrum, whether its coefficients are variable or not, soliton always propagates with variable velocity and amplitude.