scholarly journals A formula for symmetry recursion operators from non-variational symmetries of partial differential equations

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Stephen C. Anco ◽  
Bao Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Recursion Operators ◽  
Partial Differential ◽  
Variational Symmetries

scholarly journals Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

2020 ◽  
Author(s):  
Matteo Petrera ◽  
Mats Vermeeren
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integrable Hierarchies ◽  
Dimensional Space ◽  
Original System ◽  
Lagrange Equations ◽  
Partial Differential ◽  
Dimensional Space Time ◽  
Variational Symmetries ◽  
Nonlinear Math

Abstract We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in Petrera and Suris (Nonlinear Math. Phys. 24(suppl. 1):121–145, 2017) for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangian PDEs (many of which have a natural $$(1\,{+}\,1)$$ ( 1 + 1 ) -dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler–Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

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».

The geometry and invariance properties for certain classes of metrics with neutral signature

2016 ◽  
Vol 13 (06) ◽  
pp. 1650080 ◽  
Author(s):  
Jean J. H. Bashingwa ◽  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
F. D. Zaman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Algebras ◽  
Lie Group ◽  
Partial Differential ◽  
Invariance Properties ◽  
Group Methods ◽  
Variational Symmetries

In this paper, we study anti-self dual manifolds endowed with metrics of neutral signature. Since the metrics depend on solutions of, in some cases, well-known partial differential equations (PDEs), we determine exact solutions using Lie group methods. This concludes specific forms of the metrics. We then determine the isometries and the variational symmetries of the underlying metrics and corresponding Euler–Lagrange (geodesic) equations, respectively, and establish relationships between the resultant Lie algebras. In some cases, we construct conservation laws via these symmetries or the “multiplier approach”.

Partial Differential Equations

2020 ◽  
Author(s):  
A. K. Nandakumaran ◽  
P. S. Datti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential

Partial Differential Equations in Fluid Dynamics

2008 ◽  
Author(s):  
Isom H. Herron ◽  
Michael R. Foster
Keyword(s):  
Fluid Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential

Dispersive Partial Differential Equations

2016 ◽  
Author(s):  
M. Burak Erdogan ◽  
Nikolaos Tzirakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential
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