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Keywords = compacton

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15 pages, 2358 KB  
Article
Bifurcations and Exact Solutions of the Coupled Nonlinear Generalized Zakharov Equations with Anti-Cubic Nonlinearity: Dynamical System Approach
by Jie Song, Feng Li and Mingji Zhang
Mathematics 2025, 13(2), 217; https://doi.org/10.3390/math13020217 - 10 Jan 2025
Cited by 1 | Viewed by 3044
Abstract
We consider the exact traveling wave solutions for the coupled nonlinear generalized Zakharov equations. By employing the method of dynamical systems, we are able to obtain bifurcations of the phase portraits of the corresponding planar dynamical system under various parameter conditions. Based on [...] Read more.
We consider the exact traveling wave solutions for the coupled nonlinear generalized Zakharov equations. By employing the method of dynamical systems, we are able to obtain bifurcations of the phase portraits of the corresponding planar dynamical system under various parameter conditions. Based on different level curves, we derive all possible exact explicit parametric representations of bounded solutions, which include pseudo-periodic peakon, pseudo-peakon, smooth periodic wave solutions, solitary solutions, kink wave solution and the compacton solution family. Full article
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12 pages, 1090 KB  
Article
Dynamics of the Traveling Wave Solutions of Fractional Date–Jimbo–Kashiwara–Miwa Equation via Riccati–Bernoulli Sub-ODE Method through Bäcklund Transformation
by M. Mossa Al-Sawalha, Saima Noor, Mohammad Alqudah, Musaad S. Aldhabani and Roman Ullah
Fractal Fract. 2024, 8(9), 497; https://doi.org/10.3390/fractalfract8090497 - 23 Aug 2024
Cited by 2 | Viewed by 1451
Abstract
The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement [...] Read more.
The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement of the characterization of dynamic waves while providing better modeling ability compared to integer types of derivatives. The solutions of the above-mentioned time–space fractional Date–Jimbo–Kashiwara–Miwa equation have tremendous importance in numerous scientific scenarios. The regular dynamical wave solutions of the aforementioned equation encompass three fundamental functions: trigonometric, hyperbolic, and rational functions will be among the topics covered. These solutions are graphically classified into three categories: compacton kink solitary wave solutions, kink soliton wave solutions and anti-kink soliton wave solutions. In addition, to explore the impact of the fractional parameter (α) on those solutions, 2D plots are utilized, while 3D plots are applied to present the solutions involving the integer-order derivatives. Full article
(This article belongs to the Special Issue Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering)
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19 pages, 1568 KB  
Article
Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks
by Siyuan Xing and Efstathios G. Charalampidis
Entropy 2024, 26(5), 396; https://doi.org/10.3390/e26050396 - 30 Apr 2024
Cited by 1 | Viewed by 1923
Abstract
In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) [...] Read more.
In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This reformulation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as “leftons”) within the (1+1)-dimensional, b-family of PDEs. In addition, we expand our investigations, and explore not only peakon solutions in the ab-family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with multi-layer perceptron (MLP) reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs. Full article
(This article belongs to the Special Issue Recent Advances in the Theory of Nonlinear Lattices)
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33 pages, 6636 KB  
Article
Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity
by Wenjing Zhu, Zijie Ling, Yonghui Xia and Min Gao
Fractal Fract. 2023, 7(2), 201; https://doi.org/10.3390/fractalfract7020201 - 18 Feb 2023
Cited by 10 | Viewed by 2465
Abstract
This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, [...] Read more.
This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions. Full article
(This article belongs to the Topic Fractional Calculus: Theory and Applications)
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13 pages, 3815 KB  
Article
Solitons Solution of Riemann Wave Equation via Modified Exp Function Method
by Attaullah, Muhammad Shakeel, Bilal Ahmad, Nehad Ali Shah and Jae Dong Chung
Symmetry 2022, 14(12), 2574; https://doi.org/10.3390/sym14122574 - 6 Dec 2022
Cited by 22 | Viewed by 2897
Abstract
In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp [...] Read more.
In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp(−Φ(η))-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions. Full article
(This article belongs to the Special Issue Differential/Difference Equations and Its Application)
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10 pages, 2324 KB  
Article
Exact Traveling Wave Solutions in a Generalized Harry Dym Type Equation
by Rong Wu and Yan Zhou
Symmetry 2022, 14(7), 1480; https://doi.org/10.3390/sym14071480 - 20 Jul 2022
Cited by 3 | Viewed by 2498
Abstract
The traveling wave solutions of a generalized HD type equation are investigated in this study. The traveling wave system is a singular system of the first class with given parameter conditions. From the standpoint of dynamical systems, the bifurcations of traveling wave solutions [...] Read more.
The traveling wave solutions of a generalized HD type equation are investigated in this study. The traveling wave system is a singular system of the first class with given parameter conditions. From the standpoint of dynamical systems, the bifurcations of traveling wave solutions in parameter space are examined. It is demonstrated that solitary wave solutions, periodic peakons, pseudo-peakons, and compacton solutions exist. All conceivable exact explicit parametric representations of various solutions are presented. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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11 pages, 299 KB  
Article
Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients
by Lewa’ Alzaleq, Du’a Al-zaleq and Suboh Alkhushayni
Mathematics 2022, 10(5), 822; https://doi.org/10.3390/math10050822 - 4 Mar 2022
Cited by 6 | Viewed by 3003
Abstract
The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic [...] Read more.
The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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