Trilinear Form — An Extension of Hirota’s Bilinear Form

1991 ◽  
pp. 264-269 ◽  
Author(s):  
J. Satsuma ◽  
J. Matsukidaira ◽  
K. Kajiwara
Keyword(s):  
Bilinear Form ◽  
Trilinear Form ◽  
Hirota’S Bilinear Form

scholarly journals Analytical and Numerical Methods for the CMKdV-II Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Ömer Akin ◽  
Ersin Özuğurlu
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Iterative Method ◽  
Soliton Solution ◽  
Bilinear Form ◽  
Difference Method ◽  
Soliton Solutions ◽  
Hirota’S Bilinear Form ◽  
De Vries

Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) is derived. We obtain one- and two-soliton solutions analytically for the CMKdV-II. One-soliton solution of the CMKdV-II equation is obtained by using finite difference method by implementing an iterative method.

The study of complexitons and periodic solitary-wave solutions with fifth-order KdV equation in (2+1) dimensions

2019 ◽  
Vol 33 (33) ◽  
pp. 1950411 ◽  
Author(s):  
Muhammad Tahir ◽  
Aziz Ullah Awan
Keyword(s):  
Solitary Wave ◽  
Bilinear Form ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Solitary Wave Solutions ◽  
Test Technique ◽  
Wave Solutions ◽  
Complexiton Solutions ◽  
Hirota’S Bilinear Form ◽  
Fifth Order

In this paper, the generalized fifth-order (2[Formula: see text]+[Formula: see text]1)-dimensional KdV equation is scrutinized via the extended homoclinic test technique (EHTT) and extended transformed rational function (ETRF) method. With the aid of Hirota’s bilinear form, various exact solutions comprising, periodic solitary-wave, kinky-periodic solitary-wave, periodic soliton and complexiton solutions are constructed. Moreover, the mechanical features and dynamic characteristics of the obtained solutions are presented by three-dimensional plots.

Lump, periodic lump and interaction lump stripe solutions to the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (07) ◽  
pp. 1850106 ◽  
Author(s):  
Pinxia Wu ◽  
Yufeng Zhang ◽  
Iqbal Muhammad ◽  
Qiqi Yin
Keyword(s):  
Energy Distribution ◽  
Bilinear Form ◽  
Dynamic Process ◽  
Propagation Direction ◽  
Horizontal Velocity ◽  
Lump Solutions ◽  
Petviashvili Equation ◽  
Hirota’S Bilinear Form ◽  
Bkp Equation

In this paper, the Hirota’s bilinear form is employed to investigate the lump, periodic lump and interaction lump stripe solutions of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation. Many results are obtained by dynamic process of figures. We analyze the propagation direction and horizontal velocity of lump solutions to find some constraint conditions which include positiveness and localization. In the process of the travel of the periodic lump solutions, it appears that the energy distribution is not symmetrical. The interaction lump stripe solutions of non-elastic indicate that the lump solitons are dropped and swallowed by the stripe soliton.

Lump solitons and interaction phenomenon to a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

2019 ◽  
Vol 33 (32) ◽  
pp. 1950395 ◽  
Author(s):  
Na Liu ◽  
Yansheng Liu
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Nonlinear Waves ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Hirota’S Bilinear Form ◽  
Interaction Phenomenon

This paper studies lump solutions and interaction solutions for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation. With the help of symbolic computation and Hirota’s bilinear form, we obtain bright–dark lump solutions, lump-soliton solutions, and lump-kink solutions. Meanwhile, the dynamics of the obtained three classes of solutions are analyzed and exhibited mathematically and graphically. These results provide us with useful information to grasp the propagation processes of nonlinear waves.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Matroid connectivity and singularities of configuration hypersurfaces

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Graham Denham ◽  
Mathias Schulze ◽  
Uli Walther
Keyword(s):  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Smooth Locus

AbstractConsider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

Bilinear forms on non-homogeneous Sobolev spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 995-1026
Author(s):  
Carme Cascante ◽  
Joaquín M. Ortega
Keyword(s):  
Sobolev Spaces ◽  
Bilinear Form ◽  
Positive Measure ◽  
Bilinear Forms ◽  
Homogeneous Sobolev Spaces

AbstractIn this paper, we show that if {b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, {0<s<1}, by(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x}is continuous if and only if the positive measure {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for {H_{s}^{2}(\mathbb{R}^{n})}.

scholarly journals Supervised learning using a symmetric bilinear form for record linkage

2015 ◽  
Vol 26 ◽  
pp. 144-153 ◽  
Author(s):  
Daniel Abril ◽  
Vicenç Torra ◽  
Guillermo Navarro-Arribas
Keyword(s):  
Supervised Learning ◽  
Bilinear Form ◽  
Record Linkage ◽  
Symmetric Bilinear Form

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

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