New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions
TL;DR: In this article, a family of Boussinesq equations of distinct structures and dimensions are examined and the complete integrability of these equations via Painleve test is investigated.
Abstract: In the present course of study, we examine a family of Boussinesq equations of distinct structures and dimensions. We investigate the complete integrability of these equations via Painleve test. Real and complex multiple soliton solutions, for each considered model, are derived by mode of simplified Hirota’s method. Moreover, exponential expansion method has been employed to each equation, resulting into soliton solutions possessing rich spatial structure due to the presence of abundant arbitrary constants.
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TL;DR: Three bilinear auto-Backlund transformations are presented based on the Hirota method for the shallow water waves, along with some soliton solutions that depend on the water-wave coefficients in that equation.
140 citations
TL;DR: In this article, the neural network model of test function for the (3+1)-dimensional Jimbo-Miwa equation is extended to the 4-2-3 model by giving some specific activation functions.
Abstract: It is well known that most classical test functions to solve nonlinear partial differential equations can be constructed via single hidden layer neural network model by using Bilinear Neural Network Method (BNNM). In this paper, the neural network model of test function for the (3+1)-dimensional Jimbo–Miwa equation is extended to the “4-2-3” model. By giving some specific activation functions, new test function is constructed to obtain analytical solutions of the (3+1)-dimensional Jimbo–Miwa equation. Rogue wave solutions and the bright and dark solitons are obtained by giving some specific parameters. Via curve plots, three-dimensional plots, contour plots and density plots, dynamical characteristics of these waves are exhibited.
95 citations
TL;DR: In this article, the authors applied the Lie symmetry method to compute group invariant solutions for the modified Veronese web (mVw) equation and obtained its infinitesimals, commutation table of Lie algebra, symmetry reductions and closed form analytical solutions.
Abstract: The Lie symmetry method is successfully applied to compute group invariant solutions for (2 + 1)-dimensional modified Veronese web equation. The purpose of this present article is to study the modified Veronese web (mVw) equation and to obtain its infinitesimals, commutation table of Lie algebra, symmetry reductions and closed form analytical solutions. The obtained results are explicitly in the form of the functions $$f_1(y),f_2(t),f_3(x)$$ and $$f_4(x)$$ and hold numerous solitary wave solutions that are more helpful to describe dynamical phenomena through their evolution profile. The solutions are analysed physically via numerical simulation. Consequently, elastic behaviour multisolitons, line soliton, doubly soliton, parabolic wave profile, nonlinear behaviour of wave profile and elastic interaction soliton profile of solutions are demonstrated in the analysis and discussion section to make this study more praiseworthy.
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Book•
01 Jan 2004TL;DR: In this paper, Bilinearization of soliton equations is discussed and the Backlund transformation is used to transform the soliton equation into a linear combination of determinants and pfaffians.
Abstract: Preface Foreword 1. Bilinearization of soliton equations 2. Determinants and pfaffians 3. Structure of soliton equations 4. Backlund transformations Afterword References Index.
2,132 citations
TL;DR: In this paper, the authors define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equation (Burgers' equation, KdV equation, and modified KDV equation).
Abstract: In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.
1,958 citations
TL;DR: In this paper, some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented.
Abstract: Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.
922 citations
Book•
10 Sep 2010
TL;DR: Partial Differential Equations (PDE) as discussed by the authors is a family of KdV-type Equations of higher-orders, which can be found in the family of Camassa-Holm and Schrodinger Equations.
Abstract: Partial Differential Equations- Basic Concepts- First-order Partial Differential Equations- One Dimensional Heat Flow- Higher Dimensional Heat Flow- One Dimensional Wave Equation- Higher Dimensional Wave Equation- Laplace's Equation- Nonlinear Partial Differential Equations- Linear and Nonlinear Physical Models- Numerical Applications and Pade Approximants- Solitons and Compactons- Solitray Waves Theory- Solitary Waves Theory- The Family of the KdV Equations- KdV and mKdV Equations of Higher-orders- Family of KdV-type Equations- Boussinesq, Klein-Gordon and Liouville Equations- Burgers, Fisher and Related Equations- Families of Camassa-Holm and Schrodinger Equations
921 citations