scholarly journals Integrable systems of partial differential equations determined by structure equations and Lax pair

2010 ◽  
Vol 374 (4) ◽  
pp. 501-503 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integrable Systems ◽  
Lax Pair ◽  
Partial Differential ◽  
Structure Equations

scholarly journals Second-order PDEs in four dimensions with half-flat conformal structure

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190642
Author(s):  
S. Berjawi ◽  
E. V. Ferapontov ◽  
B. Kruglikov ◽  
V. Novikov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lax Pair ◽  
Conformal Structure ◽  
Second Order ◽  
Partial Differential ◽  
Characteristic Variety ◽  
Four Dimensions ◽  
Partial Classification ◽  
Monge Ampere

We study second-order partial differential equations (PDEs) in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge–Ampère property. Since half-flatness of the conformal structure is equivalent to the existence of a non-trivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge–Ampère type. Some partial classification results of Monge–Ampère equations in four dimensions with half-flat conformal structure are also obtained.

A New Reduction of the Self-Dual Yang–Mills Equations and its Applications

2016 ◽  
Vol 71 (7) ◽  
pp. 631-638 ◽  
Author(s):  
Yufeng Zhang ◽  
Yan Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Compatibility Condition ◽  
Burgers Equation ◽  
Lax Pair ◽  
Space Time ◽  
The Self ◽  
Partial Differential ◽  
Yang Mills ◽  
Modified Burgers Equation

AbstractThrough imposing on space–time symmetries, a new reduction of the self-dual Yang–Mills equations is obtained for which a Lax pair is established. By a proper exponent transformation, we transform the Lax pair to get a new Lax pair whose compatibility condition gives rise to a set of partial differential equations (PDEs). We solve such PDEs by taking different Lax matrices; we develop a new modified Burgers equation, a generalised type of Kadomtsev–Petviasgvili equation, and the Davey–Stewartson equation, which also generalise some results given by Ablowitz, Chakravarty, Kent, and Newman.

THE LAX PAIR OF A GENERALIZED THIRRING MODEL

1999 ◽  
Vol 14 (05) ◽  
pp. 659-682
Author(s):  
FABRIZIO TINEBRA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lax Pair ◽  
Thirring Model ◽  
Coupling Constants ◽  
Gauge Potential ◽  
Global Symmetry ◽  
Chiral Fermion ◽  
Partial Differential ◽  
Local Gauge Symmetry

The system of coupled nonlinear partial differential equations called the Massive Thirring Model is reviewed. In particular it is analyzed in the chiral fermion version, which is extended by introducing a local gauge symmetry in place of the usual global symmetry. This is done by minimally coupling the fermions with a SU L(2) ⊗ SU R(2) gauge potential. Following the formalism of path integrals, we then formulate the study of chiral gauge anomalies in the quantum theory of this gauge extended model. The ultimate goal of it would be to compute the divergence of the chiral fermion current for a nonlinear anomalous theory. Actually, in this work we merely exhibit the Lax pair representation of the system of partial differential equations just derived; it holds to the first order approximation in the infinitesimal gauge parameters and coupling constants present. Our aim is to hint a motivation in searching for explicit solutions of anomalous nonlinear gauge field theories, at least in simple cases.

scholarly journals A Family of Novel Exact Solutions to2+1-Dimensional KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhen Wang ◽  
Li Zou ◽  
Zhi Zong ◽  
Hongde Qin
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Lax Pair ◽  
Partial Differential ◽  
Independent Variables ◽  
New Exact Solutions

We introduce two subequations with different independent variables for constructing exact solutions to nonlinear partial differential equations. In order to illustrate the efficiency and usefulness, we apply this method to2+1-dimensional KdV equation, which was first derived by Boiti et al. (1986) using the idea of the weak Lax pair. As a result, we obtained many new exact solutions.

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