scholarly journals Negative Times of the Davey-Stewartson Integrable Hierarchy

2021 ◽  
Author(s):  
Andrei K. Pogrebkov ◽  
Keyword(s):  
Integrable Hierarchy ◽  
Lax Pairs ◽  
Negative Numbers ◽  
Integrable Equations ◽  
Lax Operator

We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.

scholarly journals Kadomtsev–Petviashvili Hierarchy: Negative Times

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1988
Author(s):  
Andrei K. Pogrebkov
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Linear Differential Equations ◽  
Lax Pairs ◽  
Negative Numbers ◽  
Infinite Hierarchy ◽  
Petviashvili Equation ◽  
Lax Operator ◽  
Leading Term ◽  
Linear Differential

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lifts up to nonlinear integrable ones by means of the special dressing procedure. Thus, one can construct not only nonlinear equations, but corresponding Lax pairs as well. The Lax operator of this evolution coincides with the Lax operator of the “positive” hierarchy. We also derive (1 + 1)-dimensional reductions of equations of this hierarchy.

scholarly journals On a method for constructing the Lax pairs for nonlinear integrable equations

2015 ◽  
Vol 49 (3) ◽  
pp. 035202 ◽  
Author(s):  
I T Habibullin ◽  
A R Khakimova ◽  
M N Poptsova
Keyword(s):  
Lax Pairs ◽  
Integrable Equations

On Generating Integrable Dynamical Systems in 1+1 and 2+1 Dimensions by Using Semisimple Lie Algebras

2015 ◽  
Vol 70 (11) ◽  
pp. 975-977 ◽  
Author(s):  
Yufeng Zhang ◽  
Honwah Tam ◽  
Lixin Wu
Keyword(s):  
Lie Algebra ◽  
Evolution Equations ◽  
Mkdv Equation ◽  
Integrable Hierarchy ◽  
Nls Equation ◽  
Soliton Equation ◽  
Integrable Equations ◽  
Zero Curvature ◽  
Integrable Dynamical Systems ◽  
Representation Method

AbstractWe deduce a set of integrable equations under the framework of zero curvature equations and obtain two sets of integrable soliton equations, which can be reduced to some new integrable equations including the generalised nonlinear Schrödinger (NLS) equation. Under the case where the isospectral functions are one-order polynomials in the parameter λ, we generate a set of rational integrable equations, which are reduced to the loop soliton equation. Under the case where the derivative λt of the spectral parameter λ is a quadratic algebraic curve in λ, we derive a set of variable-coefficient integrable equations. In addition, we discretise a pair of isospectral problems introduced through the Lie algebra given by us for which a set of new semi-discrete nonlinear equations are available; furthermore, the semi-discrete MKdV equation and the Hirota lattice equation are followed to produce, respectively. Finally, we apply the Lie algebra to introduce a set of operator Lax pairs with an operator, and then through the Tu scheme and the binomial-residue representation method proposed by us, we generate a 2+1-dimensional integrable hierarchy of evolution equations, which reduces to a generalised 2+1-dimensional Davey-Stewartson (DS) equation.

scholarly journals Integrable Nonlocal PT-Symmetric Modified Korteweg-de Vries Equations Associated with so(3, \({\mathbb{R}}\))

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2205
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Lie Algebra ◽  
Integrable Hierarchy ◽  
Integrable Equations ◽  
De Vries ◽  
Orthogonal Lie Algebra

We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra so(3,R). The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal complex reverse-spacetime and real reverse-spacetime modified Korteweg-de Vries equations associated with so(3,R).

scholarly journals ALGORITHM FOR CONSTRUCTING A LAX PAIR AND A RECURSION OPERATOR FOR INTEGRABLE EQUATIONS

2019 ◽  
Vol 47 (1) ◽  
pp. 123-126
Author(s):  
I.T. Habibullin ◽  
A.R. Khakimova
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Manifold ◽  
Nonlinear Partial Differential Equations ◽  
Recursion Operators ◽  
Lax Pairs ◽  
Partial Differential ◽  
Differential Constraint ◽  
Integrable Equations ◽  
Particular Solutions

The method of constructing particular solutions to nonlinear partial differential equations based on the notion of differential constraint (or invariant manifold) is well known in the literature, see (Yanenko, 1961; Sidorov et al., 1984). The matter of the method is to add a compatible equation to a given equation and as a rule, the compatible equation is simpler. Such technique allows one to find particular solutions to a studied equation. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) there was proposed a scheme for constructing the Lax pairs and recursion operators for integrable partial differential equations based on the use of similar idea. A suitable generalization is to impose a differential constraint not on the equation, but on its linearization. The resulting equation is referred to as a generalized invariant manifold. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) it is shown that generalized invariant varieties allow efficient construction of Lax pairs and recursion operators of integrable equations. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

Some symmetries and similarity solutions of the long-water wave hierarchy

2019 ◽  
Vol 33 (34) ◽  
pp. 1950430
Author(s):  
Xiangzhi Zhang ◽  
Yufeng Zhang ◽  
Jianqin Mei
Keyword(s):  
Conservation Laws ◽  
Water Wave ◽  
Hamiltonian Structure ◽  
Similarity Solutions ◽  
Symmetry Analysis ◽  
Integrable Hierarchy ◽  
Wave System ◽  
Invariant Solutions ◽  
Lax Pairs ◽  
Weight Method

We introduce isospectral and non-isospectral Lax pairs, then apply the Tu scheme to generate the isospectral integrable hierarchy and the non-isospectral hierarchy, whose Hamiltonian structure, hereditary operator, symmetries are followed to obtain. In addition, a kind of Bäcklund transformation of the long-water wave hierarchy for the isospectral hierarchy is constructed. Through reductions of the isospectral hierarchy, we again get the long-water wave system whose similarity solutions, nonlinear self-adjointness and the non-invariant solutions are investigated, respectively, by the use of symmetry analysis. Finally, we make use of the weight method of the variables appearing in the long-water wave system to analyze the conservation laws of the system.

scholarly journals LAX PAIRS FOR N=2,3 SUPERSYMMETRIC KdV EQUATIONS AND THEIR EXTENSIONS

1998 ◽  
Vol 13 (18) ◽  
pp. 1435-1443 ◽  
Author(s):  
S. KRIVONOS ◽  
A. PASHNEV ◽  
Z. POPOWICZ
Keyword(s):  
Infinite Family ◽  
Lax Pairs ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Lax Operator

We present the Lax operator for the N=3 KdV hierarchy and consider its extensions. We also construct a new infinite family of N=2 supersymmetric hierarchies by exhibiting the corresponding super Lax operators. The new realization of N=4 supersymmetry on the two general N=2 superfields, bosonic spin-1 and fermionic spin-1/2, is discussed.

Exploring the Boundaries of the Number–Temporal Order Association

2015 ◽  
Vol 62 (3) ◽  
pp. 198-205 ◽  
Author(s):  
Dana Ganor-Stern
Keyword(s):  
Embodied Cognition ◽  
Mental Representation ◽  
Temporal Order ◽  
Past Research ◽  
Numerical Comparison ◽  
Negative Number ◽  
Comparison Task ◽  
Negative Numbers ◽  
The Past

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1–9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

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