Wronskian solutions and Pfaffianization for a (3 <b>+</b> 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma

Cheng, Chong-Dong; Tian, Bo; Zhou, Tianyu; Shen, Yuan

doi:10.1063/5.0141559

Cited by 44 publications

(9 citation statements)

References 66 publications

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“…As we know, the wave-packet behaviours of both equations cannot be fully determined via the analytical methods, except with certain initial and boundary conditions [3,4,[7][8][9][10][11][12]. Therefore, various conventional meshbased numerical methods have been introduced to explore the Boussinesq and Camassa-Holm equations, such as the finite-element method, the finite volume method and the spectral method [20][21][22][23][24][25][26][27]. It has been revealed that for the Boussinesq equation in the 'good' case, those numerical methods can provide satisfactory results, while for the Boussinesq equation in the 'bad' case, most numerical methods fail due to the instability of the equation, even for the simulation of the simplest bell-shaped solitons [20,21].…”

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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“…It is common knowledge that nonlinear science studies chaos, solitons, and fractals [1,2]. In several areas of nonlinear physics, including hydrodynamics [3][4][5], plasmas [6][7][8], optics [9][10][11] and Bose-Einstein condensates [12][13][14][15][16][17], the study of various nonlinear waves [18][19][20][21][22][23][24][25][26] is a significant issue. Nonlinear waves include breathers, rogue waves, and solitons.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

Zhang,

Zhao,

Zhang

2024

Phys. Scr.

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In this paper, the dynamical behaviors of transformed nonlinear waves for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili and B-type Kadomtsev-Petviashvili (pKP-BKP) equation are investigated, which can be used to reveal the nonlinear wave phenomena in nonlinear optics, plasma physics and hydrodynamics. The breath-wave and the lump solutions are constructed by means of the soliton solutions. The conversion mechanism for the breath-waves is systematically analyzed, which leads to several new kink-shaped nonlinear waves. The gradient relationships of these transformed waves are revealed by a Riemannian circle. Through the analysis of the nonlinear superposition between the periodic wave component and the kink solitary wave component, the dynamical characteristics including the formation mechanism, oscillation and locality for the nonlinear waves are investigated. The time-varying properties of transformed waves are showed by the study of time variables. By virtue of the two breath-wave solution, several interactions including elastic and inelastic collisions between two nonlinear waves are studied. In particular, some transformed molecular waves encompassing the non-, semi- and full-transition modes are presented with the aid of velocity resonance. The results can help us further understand the complex nonlinear waves existing in the integrable systems.

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“…Nonlinear evolution equations (NLEEs) play an important role in describing the main feature of physical science and engineering areas, such as fluid mechanics [1,2], plasma physics [3,4], hydrodynamics [5,6], optical fibers [7][8][9][10] and chaos theory [11,12]. The solutions of these equations, especially multiwave interaction solutions, can help researchers obtain penetrating insight for some phenomena modeled by the NLEEs.…”

Section: Introductionmentioning

confidence: 99%

Multiwave interaction solutions of the partial reverse Space-time nonlocal Mel’nikov equation

Yang

Cui

2023

Phys. Scr.

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In this paper, by introducing appropriate rational and logarithmic transformations, the partial reverse Space-Time nonlocal Mel’nikov Equation in (2+1)-dimensions is transfered into its bilinear form. Then the N-soliton decomposition algorithm and the inheritance solving strategy proposed by us are extended to construct the higher order interaction solutions among solitons, periodic waves and rational waves for such type equation.

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Wronskian solutions and Pfaffianization for a (3 + 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma

Cited by 44 publications

References 66 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 transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

Multiwave interaction solutions of the partial reverse Space-time nonlocal Mel’nikov equation

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