Pfaffian, breather, and hybrid solutions for a (2 <b>+</b> 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics

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

doi:10.1063/5.0119516

Cited by 67 publications

(6 citation statements)

References 52 publications

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“…Our objective: In this paper, for equation (1), linking ρ i ʼs, we will make use of symbolic computation [29][30][31][32] to erect a Bäcklund transformation, address some solitons and present the relevant experimental support. In addition, we will employ symbolic computation to construct a family of the similarity reductions.…”

Section: ˜˜˜˜( ) ˜˜( ) ( )mentioning

confidence: 99%

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

Gao

Guo

Shan

2023

Commun. Theor. Phys.

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Recent theoretical-physics efforts have been focused on the probes for nonlinear pulse waves in the variable-radius arteries. With respect to the nonlinear waves in an artery full of blood with certain aneurysm, pulses in a blood vessel or features in a circulatory system, this paper symbolically computes out an auto-Backlund transformation via a noncharacteristic movable singular manifold, certain families of the solitonic solutions as well as a family of the similarity reductions for a variable-coefficient generalized forced-perturbed Korteweg-de Vries-Burgers equation. Aiming, e.g., at the dynamical radial displacement superimposed on the original static deformation from an arterial wall, our results rely on the axial stretch of the injured artery, blood as an incompressible Newtonian fluid, radius variation along the axial direction or aneurysmal geometry, viscosity of the fluid, thickness of the artery, mass density of the membrane material, mass density of the fluid, strain energy density of the artery, shear modulus, stretch ratio, etc. We also call the attention that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.

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Section: ˜˜˜˜( ) ˜˜( ) ( )mentioning

confidence: 99%

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

Gao

Guo

Shan

2023

Commun. Theor. Phys.

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“…As frequently used models for nonlinear processes, nonlinear evolution equations (NLEEs) have been engaged in many engineering sciences and various fields of physics, atmospheric dynamics and chemistry [1]. Many sophisticated techniques for constructing exact solutions to soliton equations have been devised, such as Bäcklund transformation [2][3][4][5], inverse scattering method [6][7][8], Darboux transformation [9][10][11][12] and Hirota bilinear method [13][14][15][16][17][18], Bernoulli expansion method [19][20][21] etc. By these methods many different types of exact solutions are obtained, such as soliton solutions [22][23][24], periodic solutions [25,26], rational solutions [27][28][29][30] and hybrid solutions [31,32], etc.…”

Section: Introductionmentioning

confidence: 99%

General high-order lump solutions and their dynamics in the Levi equations

Zhang¹,

Tang²,

Zhang³

et al. 2023

Phys. Scr.

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General high-order lump solutions are derived for the Levi equations based on the Hirota bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique. These lump solutions are given in terms of Gram determinants whose matrix elements are connected to Schur polynomials. Thus, our solutions have explicit algebraic expressions. Their dynamic behaviors are analyzed by using density maps. It is shown that when the absolute value of one group of these internal parameters in the lump solutions is very large, lump solutions exhibit obvious geometric structures. Interestingly, we have shown that their initial and middle state solutions possess various exciting geometric patterns, including hexagon, decagon, tetradecagon, etc. and other quasi-structures in addition to the standard triangle, pentagon type patterns. Because the internal parameters are not complex conjugates of each other, the dynamic behaviors of solutions are richer. These results make several contributions to the current literature and have a number of important implications for further analysis of fluid dynamics in non-homogeneous media.

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“…In recent years, more attentions have been attracted to the study of nonlinear models in optical fiber communication, plasma, condensed matter physics, and fluid mechanics [1][2][3][4]. It is well known that the Nonlinear Schrödinger (NLS) equation can be widely used in describing the quantum behavior of microscopic particles in quantum mechanics [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

The cell-average based neural network for numerical approximation of the nonlinear Schrödinger equation

Hu,

Qiu,

Yang

et al. 2023

Preprint

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Given the advantages of machine learning method with accuracy and high efficiency, when compared with classical numerical methods, the cell-average based neural network (CANN) method is proposed to opens up a new field for solving and simulating solutions of nonlinear Schrödinger equations numerically. Inspired by the finite volume method for solving fluid flow problems, the CANN method seeks to explore shallow and fast neural network solvers to approximate the solution average difference between two consecutive time levels. The CANN method can be considered as a time discretization scheme, which is an explicit one-step method that evolves the solution forward in time. The CANN method is a network method of the finite volume type, which means that it is mesh dependent and is a local solver. The experimental results demonstrate that the CANN method yields satisfactory outcomes at different time steps within the specified time window. Once the neural network has been effectively trained, it can be applied to solve the NLS equation with different initial conditions. Furthermore, the CANN method demonstrates strong generalization ability in processing low-quality data with noise. To enhance the utilization of the CANN method in partial differential equations, we carry out numerical experiments on the NLS equation, which show the high practicality and accuracy of this method.

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Pfaffian, breather, and hybrid solutions for a (2 + 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics

Cited by 67 publications

References 52 publications

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

General high-order lump solutions and their dynamics in the Levi equations

The cell-average based neural network for numerical approximation of the nonlinear Schrödinger equation

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