scholarly journals Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

2011 ◽  
Vol 66 (6-7) ◽  
pp. 377-382 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Aslı Pekcan
Keyword(s):  
Natural Number ◽  
Soliton Solution ◽  
Boussinesq Equations ◽  
Uniqueness Property ◽  
Integrable Equations ◽  
Completely Integrable ◽  
Arbitrary Natural Number ◽  
Integrability Theory

The Kadomtsev-Petviashvili and Boussinesq equations (uxxx -6uux)x -utx ±uyy = 0; (uxxx - 6uux)x +uxx ±utt = 0; are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. It is shown that the Kadomtsev-Petviashvili and Boussinesq equations and their dimensional reductions are the only integrable equations among a class of generalized Kadomtsev-Petviashvili and Boussinesq equations (ux1x1x1 - 6uux1 )x1 + ΣMi;j=1aijuxixj = 0; where the aij’s are arbitrary constants and M is an arbitrary natural number, if the existence of the three-soliton solution is required

A Two-dimensional Boussinesq equation for water waves and some of its solutions

1996 ◽  
Vol 323 ◽  
pp. 65-78 ◽  
Author(s):  
R. S. Johnson
Keyword(s):  
Soliton Solution ◽  
Water Waves ◽  
Boussinesq Equation ◽  
Kdv Equation ◽  
Boussinesq Equations ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Completely Integrable ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

scholarly journals New Interaction Solutions of (3+1)-Dimensional KP and (2+1)-Dimensional Boussinesq Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Bo Ren ◽  
Jun Yu ◽  
Xi-Zhong Liu
Keyword(s):  
Soliton Solution ◽  
Boussinesq Equations ◽  
Phase Shifts ◽  
Kp Equation ◽  
Cnoidal Waves ◽  
Interaction Solution ◽  
Interaction Solutions

The consistent tanh expansion (CTE) method has been succeeded to apply to the nonintegrable (3+1)-dimensional Kadomtsev-Petviashvili (KP) and (2+1)-dimensional Boussinesq equations. The interaction solution between one soliton and one resonant soliton solution for the (3+1)-dimensional KP equation is obtained with CTE method. The interaction solutions among one soliton and cnoidal waves for these two equations are also explicitly given. These interaction solutions are investigated in both analytical and graphical ways. It demonstrates that the interactions between one soliton and cnoidal waves are elastic with phase shifts.

scholarly journals Quantum Multiple Construction of Subfactors

2008 ◽  
Vol 51 (3) ◽  
pp. 321-333
Author(s):  
Marta Asaeda
Keyword(s):  
Natural Number ◽  
Finite Index ◽  
Tensor Category ◽  
Quantum Double ◽  
Arbitrary Natural Number

AbstractWe construct the quantum s-tuple subfactors for an AFD II1 subfactor with finite index and depth, for an arbitrary natural number s. This is a generalization of the quantum multiple subfactors by Erlijman and Wenzl, which in turn generalized the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.

scholarly journals Permutations that preserve asymptotically null sets and statistical convergence

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4821-4827
Author(s):  
Jeff Connor
Keyword(s):  
Natural Number ◽  
Statistical Convergence ◽  
Asymptotic Density ◽  
Arbitrary Natural Number ◽  
Convergent Sequences

The main result of this article is a characterization of the permutations ?: N ? N that map a set with zero asymptotic density into a set with zero asymptotic density; a permutation has this property if and only if the lower asymptotic density of Cp tends to 1 as p ? ? where p is an arbitrary natural number and Cp = {l : ?-1(l)? lp}. We then show that a permutation has this property if and only if it maps statistically convergent sequences into statistically convergent sequences.

scholarly journals On Orbits of the Ring Zmn under Action of the Group SL(m, Zn)

10.14311/768 ◽  
2005 ◽  
Vol 45 (5) ◽  
Author(s):  
P. Novotný ◽  
J. Hrivnák
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Matrix Group ◽  
Arbitrary Natural Number ◽  
Finite Matrix

We consider the action of the finite matrix group SL(m,Zn ) on the ring Zmn. We determine orbits of this action for n arbitrary natural number. It is a generalization of the task which was studied by A. A. Kirillov for m = 2 and n prime number.

The theory of Legendrian unfoldings and first-order differential equations

1993 ◽  
Vol 123 (3) ◽  
pp. 517-532 ◽  
Author(s):  
Shyuichi Izumiya
Keyword(s):  
Differential Equations ◽  
Singular Solutions ◽  
Integrable Equations ◽  
First Order ◽  
Completely Integrable ◽  
Open Set ◽  
Complete Solutions ◽  
First Order Differential Equations

SynopsisWe consider some properties of completely integrable first-order differential equations for real-valued functions. In order to study this subject, we introduce the theory of Legendrian unfoldings. We give a characterisation of equations with classical complete solutions in terms of Legendrian unfoldings, and also assert that the set of equations with singular solutions is an open set in the space of completely integrable equations even though such a set is thin in the space of all equations.

A remark on abstract operator completely integrable equations

1980 ◽  
Vol 80 (4) ◽  
pp. 213-216 ◽  
Author(s):  
D.V. Chudnovsky ◽  
G.V. Chudnovsky
Keyword(s):  
Integrable Equations ◽  
Completely Integrable ◽  
Abstract Operator
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