Potential symmetries and conservation laws for generalized quasilinear hyperbolic equations

Transformations of quasilinear systems originating from the projective theory of congruences

We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the Lévy transformations of semihamiltonian systems. Correspondence between commuting flows and certain families of planes containing the lines of the congruence is pointed out. In the particular case n=2 this construction provides an explicit parametrization of surfaces, harmonic to…

Killing fields and Conservation Laws for rank-1 Toda field equations

We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzeica equation, completing the work in Fox and Goertsches (2011). We define a notion of finite-type for integral manifolds of exterior differential systems directly in terms of conservation laws that generalizes the definition of Pinkall-Sterling (1989). The definition applies to an…

Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians

Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the authors showed the mathematical structure behind both the Euler-Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. In this paper we demonstrate that the knowledge of this structure considerably eases finding the extremal curves for variational problems inva…

Higher-order conservation laws for the non-linear Poisson equation via the characteristic cohomology

We study higher-order conservation laws of the non-linearizable elliptic Poisson equation $ \frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u) $ as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. The S^1--symmetry of the PDE leads to a normal form for the undifferentiated conservation law as well. We show that for higher-order conservation laws to exist, it is necess…

Construction of Local Conservation Laws by Generalized Isometric Embeddings of Vector Bundles

This article uses Cartan-Kähler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free energy-momentum tensors. We also generalize the local isometric embedding of surfaces in the analytic case by applying the main result to vector bundles of rank two over any surface.

关于多重调和映射守恒律的研究

On conservation laws for polyharmonic maps

本文综述了黎曼流形间多重调和映射的守恒律，揭示了能量变分导出的应力-能量张量及目标流形Killing向量场导出的守恒律，并指出其在调和与双调和映射中的应用。