arXiv subjects

Ian M. Anderson

Publications and source records attributed to Ian M. Anderson.

5 recordsLinked to original sources

arXiv2015

Backlund Transformations for Darboux Integrable Differential Systems: Examples and Applications

In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of Bäcklund transformations. We then showed how this construction can be applied to construct Bäcklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new applications which demonstrate the theory. In particular our final example demonstrates how our group theoretical approach produces all the Bäcklund transformations in arXiv:0707.4408. We also prove, using group methods, a non-existence theorem for Bäcklund transformatons which disagrees with part 2 of Theorem 1 in arXiv:0707.4408.

math.DG↗

arXiv2015

Explicit ambient metrics and holonomy

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic rank 2 and 3 distributions in respective dimensions 5 and 6. The corresponding explicit Fefferman-Graham ambient metrics provide a large class of metrics with holonomy equal to the exceptional non-compact Lie group $\mathbf{G}_2$ as well as ambient metrics with holonomy contained in $\mathbf{Spin}(4,3)$.

math.DG↗

arXiv2014

Backlund Transformations for Darboux Integrable Differential Systems

We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.

math.DG↗

arXiv2013

The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas

To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the Vessiot group $G$. If the Vessiot group $G$ is solvable then the Cauchy problem can be solved by quadratures. This allows us to give explicit integral formulas, similar to the well known d'Alembert's formula for the wave equation, to the initial value problem with generic non-characteristic initial data.

math.DG↗

arXiv2012

Variational Principles for Natural Divergence-free Tensors in Metric Field Theories

Let $T^{ab}=T^{ba}=0$ be a system of differential equations for the components of a metric tensor on $R^m$. Suppose that $T^{ab}$ transforms tensorially under the action of the diffeomorphism group on metrics and that the covariant divergence of $T^{ab}$ vanishes. We then prove that $T^{ab}$ is the Euler-Lagrange expression some Lagrangian density provided that $T^{ab}$ is of third order. Our result extends the classical works of Cartan, Weyl, Vermeil, Lovelock, and Takens on identifying field equations for the metric tensor with the symmetries and conservation laws of the Einstein equations.

math-ph↗