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Bäcklund Transformations and Zero‐Curvature Representations of Systems of Partial Differential Equations [PDF]

Journal of Mathematical Physics, 1994
It is shown that Bäcklund transformations (BTs) and zero-curvature representations (ZCRs) of systems of partial differential equations (PDEs) are closely related. The connection is established by nonlinear representations of the symmetry group underlying the ZCR which induce gauge transformations relating different BTs.
Friedemann Brandt
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"ISOMONODROMY" SOLUTIONS OF EQUATIONS OF ZERO CURVATURE

Mathematics of the USSR-Izvestiya, 1986
See the review in Zbl 0589.35087.
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τ Functions and Zero Curvature Equations of Toda–AKNS Type

Journal of Mathematical Physics, 1988
The connection between τ functions and zero curvature equations for the homogeneous construction of the basic module L(Λ0) over the simplest affine Kac–Moody algebra A(1)1 is studied.
Bergvelt, M. J., ten Kroode, A. P. E.
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The zero curvature equation for rigid CR-manifolds

Complex Variables and Elliptic Equations, 2015
In this paper, we present the general analytic solution to the zero curvature equation for rigid three-dimensional CR-manifolds. The solutions are uniquely determined by one function and four real parameters.
V. Ezhov, G. Schmalz
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A Generating Scheme for Conservation Laws of Discrete Zero Curvature Equations and Its Application

Computers and Mathematics With Applications, 2019
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Wen-Xiu Ma
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On Liouville Integrability of Zero-Curvature Equations and the Yang Hierarchy

Journal of Physics A, 1989
Sufficient conditions for a zero-curvature equation Ut-Vx+(U,V)=0 being Liouville integrable are investigated. In the case that the equation is integrable an explicit formula of the Poisson bracket (H( lambda ),H( mu )) for Hamiltonians H is proposed. The Yang hierarchy is derived and shown to be Liouville integrable.
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On a method for constructing algebro-geometric solutions to the zero curvature equation

Theoretical and Mathematical Physics, 1997
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Novikov, D. P., Romanovskij, R. K.
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Liouville Integrability of Zero Curvature Equations

1990
Let S = S(∞, −∞) be the Schwartz space, and $${u_t} = K(u)$$ (1) be a nonlinear evolution equation (NLEE), where u = (u l,..., u p ) ∈ S p . There are different definitions on integrability of the equation (1), we shall adopt the following two definitions: (A.) We call the equation (1) Lax integrable if it can be written as a zero ...
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On conservation laws and zero-curvature representations of the Liouville equation

Journal of Physics A: Mathematical and General, 1994
Summary: Applying the first Noether theorem to the Liouville equation \(u_{xy}= \exp u\), we find all (namely, a continuum of) non-trivial conservation laws of this equation. Then we find five new zero-curvature representations of the Liouville equation (by \(2\times 2\) traceless matrices) which contain, respectively, 1, 1, 2, 2 and 3 essential ...
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A new hyperbolic equation possessing a zero-curvature representation

Journal of Mathematical Sciences, 2006
The author found a certain new example of a hyperbolic nonlinear equation \(u_{xy}=F(x,y,u,u_x,u_y)\) of the zero-curvature representation with values in \(sl_2\). It is proved that this representation does not contain parameters and can not be reduced to the zero-curvature representation inside proper subalgebras.
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