Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrödinger equation

Rizvi, Syed T. R.; Seadawy, Aly R.; Ahmed, Sarfaraz; Younis, Muhammad; Ali, Kashif

doi:10.1016/j.chaos.2021.111251

Cited by 125 publications

(23 citation statements)

References 60 publications

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“…For the Boussinesq equation, the solitons are bellshaped, and for the Camassa-Holm equation, when κ → 0, the solitons propagate with sharp peaks, also known as peakons [3,4,[9][10][11]. The analytical soliton and rogue wave solutions can be obtained through the direct method or inverse scattering transformation [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. For example, with the direct Hirota's method, the Boussinesq equation can be transformed as…”

Section: Introductionmentioning

confidence: 99%

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

Su,

Zhang,

Lan

et al. 2024

Proc. R. Soc. A.

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The Boussinesq and Camassa–Holm equations are, respectively, used to describe the bidirectional and unidirectional motions of small-amplitude waves on shallow water surfaces, but their wave dynamics remain a challenge for almost all conventional numerical methods. In this paper, we use physics-informed neural networks (PINNs), a mesh-free deep learning method, to accurately predict the soliton (peakon) interaction or rogue wave behaviours of both equations with only a few initial and boundary data, providing a method for solving certain numerically unstable fluid systems or extreme wave solutions of regular fluid systems. It is revealed that by decomposing both of the equations into lower-order coupled systems, one can obtain higher-precision wave behaviours, especially for the Camassa–Holm equation. To retrieve certain unknown hydraulic parameters based on the observed wave data, e.g. the water depth, we use a multiple PINN method and stably identify the dynamic parameters in the Boussinesq equation through the association of localized soliton and rogue wave solutions. Furthermore, we compare the PINNs with conventional high-precision time-splitting Fourier spectral (TSFS) method and find that to achieve the same split feature of the Y-shapedsoliton of the Boussinesq equation, PINNs require only one-third of the initial and boundary data of the TSFS method.

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Section: Introductionmentioning

confidence: 99%

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

Su,

Zhang,

Lan

et al. 2024

Proc. R. Soc. A.

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“…In recent decades, fractional calculus has been a significant mathematical technique used to represent critical challenges in various areas, including science, technology, and engineering such as, optimal power flow problems 17 , nonlinear output-error systems 18 , recommender systems with chaotic ratings behavior 19 , parameter estimation of nonlinear control autoregressive systems 20 , power management involving wind-load chaos and uncertainties 21 , Schrodinger equations 22 , 23 and shallow water waves 24 . The power-law function is involved in the Liouville–Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of chaotic system based on circuit design with Ulam stability through fractal-fractional derivative with power law kernel

Khan

Ahmad

Shah

et al. 2023

Sci Rep

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In this paper, the newly developed Fractal-Fractional derivative with power law kernel is used to analyse the dynamics of chaotic system based on a circuit design. The problem is modelled in terms of classical order nonlinear, coupled ordinary differential equations which is then generalized through Fractal-Fractional derivative with power law kernel. Furthermore, several theoretical analyses such as model equilibria, existence, uniqueness, and Ulam stability of the system have been calculated. The highly non-linear fractal-fractional order system is then analyzed through a numerical technique using the MATLAB software. The graphical solutions are portrayed in two dimensional graphs and three dimensional phase portraits and explained in detail in the discussion section while some concluding remarks have been drawn from the current study. It is worth noting that fractal-fractional differential operators can fastly converge the dynamics of chaotic system to its static equilibrium by adjusting the fractal and fractional parameters.

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“…Physics and engineering are two mathematics-based sciences that frequently use PDEs [1][2][3][4]. The fundamentals of contemporary scientific logic are formed by them for several ideas, including heat, sound, diffusion, electrodynamics [5][6][7], electrostatics, elastic, hydrodynamics, and quantum mechanics [8][9][10]. A wide range of scientific fields are interested in studying nonlinear wave phenomena [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

2023

Self Cite

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We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

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Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrödinger equation

Cited by 125 publications

References 60 publications

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

Dynamics of chaotic system based on circuit design with Ulam stability through fractal-fractional derivative with power law kernel

Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

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