A. Michel Grundland
To establish a relation between two approaches to the construction of Riemann k-wave solutions of hydrodynamic-type systems, namely the symmetry reduction method and the generalized method of characteristics.
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We improve the concept of our previous paper "Dirac type tensor equations with nonabelian gauge symmetries on pseudo-Riemannian space" and present a new compact formula for the tensor $B_μ$.
We suggest a so-called Dirac type tensor equation with nonabelian gauge symmetry on pseudo-Riemannian space. This equation reproduce some of the properties of spinor Dirac equation. A geometrical interpretation of results in terms of Riemannian geometry is given.
In this paper it is shown that Riemann invariants are invariant under nonclassical symmetries of a hyperbolic system. As a specific example, we study the one-dimensional shallow water equations on the flat and present another type invariance under nonclassical symmetries.
This paper contains an analysis of rank-k solutions in terms of Riemann invariants, obtained from interrelations between two concepts, that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of PDEs in many dimensions. A variant of the conditional symmetry method for obtaining this type of solutions is proposed. A Lie module of vector fields, which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints, is constructed. It is shown that th…
We study the reduction of the heat equation in Riemannian spaces which admit a gradient Killing vector, a gradient homothetic vector and in Petrov Type D,N,II and Type III space-times. In each reduction we identify the source of the Type II hidden symmetries. More specifically we find that a) If we reduce the heat equation by the symmetries generated by the gradient KV the reduced equation is a linear heat equation in the nondecomposable space. b) If we reduce the heat equation via the symmetries generated by the gradient HV the reduced equation is a Laplace equation for an appropriate metric.…
We prove a general theorem which allows the determination of Lie symmetries of Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. In each reduction we identify the source of Type II hidden symmetries. We find that in general the Type II hidden symmetries of Laplace equation are directly related to the tran…
A new approach to the solution of quasilinear nonelliptic first-order systems of inhomogeneous PDEs in many dimensions is presented. It is based on a version of the conditional symmetry and Riemann invariant methods. We discuss in detail the necessary and sufficient conditions for the existence of rank-2 and rank-3 solutions expressible in terms of Riemann invariants. We perform the analysis using the Cayley-Hamilton theorem for a certain algebraic system associated with the initial system. The problem of finding such solutions has been reduced to expanding a set of trace conditions on wave ve…
Symmetries and the corresponding fields of differential invariants of the inviscid flows on a curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given.
Ermakov systems possessing Noether point symmetry are identified among the Ermakov systems that derive from a Lagrangian formalism and, the Ermakov invariant is shown to result from an associated symmetry of dynamical character. The Ermakov invariant and the associated Noether invariant, are sufficient to reduce these systems to quadratures.
Nonlinear PDE's having {\bf given} conditional symmetries are constructed. They are obtained starting from the invariants of the "conditional symmetry" generator and imposing the extra condition given by the characteristic of the symmetry. Several of examples starting from the Boussinesq and including non-autonomous Korteweg-De Vries like equations are given to showcase the methodology introduced.
Using the basic prolongation method and the infinitesimal criterion of invariance, we find the most general Lie point symmetries group of the Thomas equation. Looking the adjoint representation of the obtained symmetry group on its Lie algebra, we will find the preliminary classification of its group-invariant solutions. This latter provides a new exact solutions for the Thomas equation.
74J30The maximal group of Lie point symmetries of a system of nonlinear equations used in geophysical fluid dynamics is presented. The Lie algebra of this group is infinite-dimensional and involves three arbitrary functions of time. The invariant solution under the rotation and dilation is constructed. Qualitative analysis of the invariant solution is provided and the energy of this solution is presented.
The generalized method of characteristics is used to obtain rank-2 solutions of the classical equations of hydrodynamics in (3+1) dimensions describing the motion of a fluid medium in the presence of gravitational and Coriolis forces. We determine the necessary and sufficient conditions which guarantee the existence of solutions expressed in terms of Riemann invariants for an inhomogeneous quasilinear system of partial differential equations. The paper contains a detailed exposition of the theory of simple wave solutions and a presentation of the main tool used to study the Cauchy problem. A s…
In this paper we employ a "direct method" in order to obtain rank-k solutions of any hyperbolic system of first order quasilinear differential equations in many dimensions. We discuss in detail the necessary and sufficient conditions for existence of these type of solutions written in terms of Riemann invariants. The most important characteristic of this approach is the introduction of specific first order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method of solving multi-dimensional systems of PDEs. We have demonstrated th…
Existence of Riemannian invariants for integrable systems of hydrodynamic type
本文研究了可积流体类型系统中黎曼不变量的存在性，证明在特定坐标系下系统生成元和对称性可对角化。
Inspired by recent formulæ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov-Witten invariants of $\mathbb{P}^1$ as an isomonodromic tau function associated with a difference equation. As a byproduct we obtain an analogue of the Kontsevich matrix model for this tau function. A connection with the Charlier ensemble is also considered.
Differential constraints for hyperbolic systems through k-Riemann invariants
本文基于微分约束法和 k-Riemann 不变量，提出求解一维非齐次拟线性双曲系统精确波解的约化方法，并应用于带源项 Euler 系统以刻画稀疏波和求解 Riemann 问题。
The different natures of approximate symmetries and their corresponding first integrals/invariants are delineated in the contexts of both Lie symmetries of ordinary differential equations and Noether symmetries of the Action Integral. Particular note is taken of the effect of taking higher orders of the perturbation parameter. Approximate symmetries of approximate first integrals/invariants and the problems of calculating them using the Lie method are considered.
We extend the ideas of Kolmogorov theory on symmetries of turbulent dynamics to analyze invariants, scaling and spectra of unsteady turbulent mixing induced by the Rayleigh-Taylor instability. Time- and scale-invariance of the rate of momentum loss leads to non-dissipative momentum transfer between the scales, to and power-law scale-dependencies of the velocity and Reynolds number and spectra distinct from Kolmogorov, and sets the viscous and dissipation scales. Turbulent mixing exhibits more order compared to isotropic turbulence. Mechanisms of flow regularization are discussed.
Solid state systems with time reversal symmetry and/or particle-hole symmetry often only have $\mathbb{Z}_2$-valued strong invariants for which no general local formula is known. For physically relevant values of the parameters, there may exist approximate symmetries or almost conserved observables, such as the spin in a quantum spin Hall system with small Rashba coupling. It is shown in a general setting how this allows to define robust integer-valued strong invariants stemming from the complex theory, such as the spin Chern numbers, which modulo $2$ are equal to the $\mathbb{Z}_2$-invariants…
In this letter we study a class of symmetries of the new translational extended shape invariant potentials. It is proved that a generalization of a compatibility condition introduced in a previous article is equivalent to the usual shape invariance condition. We focus on the recent examples of Odake and Sasaki (infinitely many polynomial, continuous $l$ and multi-index rational extensions). As a byproduct, we obtain new relations, to the best of our knowledge, for Laguerre, Jacobi polynomials and (confluent) hypergeometric functions.
We give a systematic local description of invariant metrics and other invariant fields on a spacetime under the action of a (non-abelian) group. This includes the invariant fields in a neighbourhood of a principal and a special orbit. The construction is illustrated with examples. We also apply the formalism to give the R-symmetry invariant metrics of some AdS backgrounds and comment on applications to Kaluza-Klein theory.
We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them for both equations. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly met…
We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schrödinger equations.
Following up the work of [1] on deformed algebras, we present a class of Poincaré invariant quantum field theories with particles having deformed internal symmetries. The twisted quantum fields discussed in this work satisfy commutation relations different from the usual bosonic/fermionic commutation relations. Such twisted fields by construction are nonlocal in nature. Despite this nonlocality we show that it is possible to construct local interaction Hamiltonians which satisfy cluster decomposition principle and are Lorentz invariant. We further illustrate these ideas by considering global S…
Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular invariants, we determine the algebras of quantum symmetries, obtain their generators, and, as a by-product, recover the known graphs E4(SU4), E6(SU4) and E8(SU4) describing exceptional quantum subgroups of type SU(4). We also obtain characteristic numbers (quantum cardinalities, dimensions) for each of them and for their associated quantum groupoids.
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