Isospectral Pair Research Articles

Non-Compact Inaudibility of Weak Symmetry and Commutativity via Generalized Heisenberg Groups

Two Riemannian manifolds are said to be isospectral if there exists a unitary operator which intertwines their Laplace-Beltrami operator. In this paper, we prove in the non-compact setting the inaudibility of the weak symmetry property and the commutative property using an isospectral pair of 23 dimensional generalized Heisenberg groups.

Geometric structures and the Laplace spectrum, Part II

We classify the locally homogeneous Riemannian metrics on elliptic three-manifolds, which contributes to our classification of the compact homogeneous Riemannian three-manifolds. Then, we apply the former result to our continued exploration of the extent to which the spectrum encodes the local geometry of a locally homogeneous three-manifold. Specifically, using the first four heat invariants, we find that any collection of isospectral locally homogeneous metrics on an elliptic three-manifold Γ ∖ S 3 \Gamma \backslash S^3 contains at most two isometry classes and these classes are necessarily locally isometric. In fact, if the elliptic three-manifold is S 3 S^3 , R P 3 \mathbb {R}P^3 or has non-cyclic fundamental group, then (up to isometry) its locally homogeneous Riemannian metrics can be mutually distinguished via their spectra. Currently, there is no example of an isospectral pair consisting of locally homogeneous elliptic three-manifolds with non-isometric universal cover; however, we show that if such a pair exists, then it satisfies certain restrictive geometric conditions. Finally, we note that our classification of locally homogeneous elliptic three-manifolds shows that, for q ≥ 3 q \geq 3 , the lens space L ( q ; 1 , 1 ) L(q;1,1) admits pairs of locally isometric locally homogeneous metrics where only one of the metrics is homogeneous. While this phenomenon cannot occur in dimension two, these lens spaces account for all such examples in dimension three.

Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation

We study the Veronese web equation uyutx+λuxuty−(λ+1)utuxy=0 and using its isospectral Lax pair construct two infinite series of nonlocal conservation laws. In the infinite differential coverings associated to these series, we describe the Lie algebras of the corresponding nonlocal symmetries. Finally, we construct a recursion operator and explore its action on nonlocal shadows. The operator provides a new shadow which serves as a master-symmetry.

The generalized Giachetti-Johnson hierarchy and algebro-geometric solutions of the coupled KdV-MKdV equation

By using a Lie algebra A1, an isospectral Lax pair is introduced from which a generalized Giachetti-Johnson hierarchy is generated, which reduce to the coupled KdV-MKdV equation, furthermore, the algebro-geometric solutions of the coupled KdV-MKdV equation are constructed in terms of Riemann theta functions.

Non-isometric Riemannian G-Manifolds with Equal Equivariant Spectra

In this paper, the author examines the two methods that people used to systematically construct isospectral non-isometric Riemannian manifolds, the Sunada–Pesce–Sutton method and the torus action method, and shows that both methods can be used to produce equivariantly isospectral non-isometric Riemannian G-manifolds. The author also shows that the Milnor’s isospectral pair is not equivariantly isospectral.

Algebro-Geometric Solutions with Characteristics of a Nonlinear Partial Differential Equation with Three-Potential Functions* *Supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014) and the National Natural Science Foundation of China under Grant No. 11371361, the Fundamental Research Funds for the Central Universities (2013XK03) as well as the Natural Science Foundation of Shandong

With the help of a simple Lie algebra, an isospectral Lax pair, whose feature presents decomposition of element (1, 2) into a linear combination in the temporal Lax matrix, is introduced for which a new integrable hierarchy of evolution equations is obtained, whose Hamiltonian structure is also derived from the trace identity in which contains a constant γ to be determined. In the paper, we obtain a general formula for computing the constant γ. The reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions, the mKdV equation and a generalized linear KdV equation. The algebro-geometric solutions (also called finite band solutions) of the generalized nonlinear heat equation are obtained by the use of theory on algebraic curves. Finally, two kinds of gauge transformations of the spatial isospectral problem are produced.

On generating linear and nonlinear integrable systems with variable coefficients

Under an isospectral Lax pair, a new integrable hierarchy of evolution equations is obtained by starting from a given Lie algebra T, which can be reduce to a new coupled integrable equation similar to the long wave equation, but it is not the standard long wave equation. By making use of an enlarged Lie algebra T1 of the Lie algebra T, we obtain a type of equation hierarchy (called a linear hierarchy). The corresponding Hamiltonian structure of the equation hierarchy is derived from the variational identity. As we all know that nonlinear equations with variable coefficients can be used to describe some real phenomena in physical and engineering fields. It is an interesting and important topic to consider how to generate variable-coefficient nonlinear integrable equations from the mathematical viewpoint. In the paper, we construct another enlarged Lie algebra T2 of the Lie algebra T for which an integrable hierarchy (called the nonlinear hierarchy) of nonlinear integrable equations with variable coefficients is obtained. Furthermore, the Hamiltonian structure of the integrable hierarchy is produced by using the variational identity again. As long as the linear hierarchy and the nonlinear hierarchy are derived, following their reductions, some linear and nonlinear evolution equations with variable coefficients are obtained, respectively. The corresponding Hamiltonian structures of such reduced equations with variable coefficients are followed to present.

GENERALIZED mKdV EQUATION, LIOUVILLE EQUATION, SINE-GORDON EQUATION AND SINH-GORDON EQUATION AS WELL AS A FORMAL BÄCKLUND TRANSFORMATION

A Lie algebra which consists of linear combinations of one basis of the Lie algebra A1is presented for which an isospectral Lax pair is exhibited. By using the zero curvature equation, the generalized mKdV equation, Liouville equation and sine-Gordon equation, sinh-Gordon equation are generated via polynomial expansions. Finally, we investigate a kind of formal Bäcklund transformation for the generalized sine-Gordon equation. The explicit Bäcklund transformation of the standard sine-Gordon equation is presented. The other equations given in the paper are obtained similarly.

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy

A Lie algebra sl(2) which is isomorphic to the known Lie algebra A 1 is introduced for which an isospectral Lax pair is presented, whose compatibility condition leads to a soliton-equation hierarchy. By using the trace identity, its Hamiltonian structure is obtained. Especially, as its reduction cases, a Sine equation and a complex modified KdV(cmKdV) equation are obtained,respectively. Then we enlarge the sl(2) into a bigger Lie algebra sl(4) so that a type of expanding integrable model of the hierarchy is worked out. However, the soliton-equation hierarchy is not integrable couplings. In order to generate the integrable couplings, an isospectral Lax pair is introduced. Under the frame of the zero curvature equation, we generate an integrable coupling whose quasi-Hamiltonian function is derived by employing the variational identity. Finally, two types of computing formulas of the constant γ are obtained, respectively.

A New Liouville Integrable Hamiltonian System

With the help of a Lie algebra, an isospectral Lax pair is introduced for which a new Liouville integrable hierarchy of evolution equations is generated. Its Hamiltonian structure is also worked out by use of the quadratic-form identity.

A NEW EXPANDING INTEGRABLE HIERARCHY OF KAUP–NEWELL HIERARCHY

A higher loop algebra is constructed which is devoted to establish an isospectral Lax pair. Integrable coupling of the well-known Kaup–Newell hierarchy is obtained by using a relation of direct sum between two sub-algebras. As in the reduction case, an integrable coupling of a generalized MKdV equation is presented. The approach presented in the paper can be used generally.

A higher-dimensional Lie algebra and its decomposed subalgebras

A higher-dimensional 4×4 matrix Lie algebra G is presented, which is devoted to setting up an isospectral Lax pair whose compatibility generates an integrable coupling system. By employing the quadratic-form identity, its Hamiltonian structure is obtained. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra G. If E is decomposed into two subalgebras E1 and E2 and we require the closure between E1 and E2 under the matrix multiplication, that is, E1E2, E2E1⊂E2, then a discrete lattice integrable coupling system is worked out. A remarkable feature of the Lie algebras G and E is used to directly construct integrable couplings.

Equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains

Isospectrality of the planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to isolength spectrality of the corresponding domains. Although an explicit and exact trace formula such as Poisson's summation formula or Selberg's trace formula is not known to exist for such planar domains, equivalence between isospectrality and isolength spectrality in a certain setting can be proved by employing the matrix representation of `transplantation of eigenfunctions'. As an application of the equivalence, transplantable pairs of domains, which are all isospectral pair of planar domains and therefore counter-examples of Kac's question `can one hear the shape of a drum?', are numerically enumerated, and it is found that, at least up to the domain composed of 13 building blocks, transplantable pairs coincide with those constructed by the method due to Sunada.

Self-isospectral periodic potentials and supersymmetric quantum mechanics

We discuss supersymmetric quantum mechanical models with periodic potentials. The important new feature is that it is possible for both isospectral potentials to support zero modes, in contrast with the standard nonperiodic case where either one or neither (but not both) of the isospectral pair has a zero mode. Thus it is possible to have supersymmetry unbroken and yet also have a vanishing Witten index. We present some explicit exactly soluble examples for which the isospectral potentials have identical band spectra, and which are ``self-isospectral'' in the sense that the potentials have identical shape, but are translated by one half-period relative to one another.

On irreducible endospectral graphs

The paper considers endospectral trees, a special class of graphs associated with the production of numerous isospectral graphs. Endospectral graphs have been considered in the literature sporadically (the name was suggested very recently [M. Randić, SIAM J. Algebraic Discrete Meth. 6, 145 (1985)]). They are characterized by the presence of a pair of special vertices that, if replaced by any fragment, produce an isospectral pair of graphs. Recently Jiang [Y. Jiang, Sci. Sin. 27, 236 (1984)] and Randić and Kleiner (M. Randić and A. F. Kleiner, ‘‘On the construction of endospectral trees,’’ submitted to Ann. NY Acad. Sci.) considered alternative constructive approaches to endospectral trees and listed numerous such graphs. The listing of all such trees having n=16 or fewer vertices has been undertaken here. It has been found that relatively few endospectral trees have novel structural features and cannot be reduced to some already known endospectral tree. These few have been named ‘‘irreducible endospectral trees.’’ They are responsible for the occurrence of a large number of isospectral trees, leading to, when one considers trees of increasing size, the situation that led Schwenk [A. J. Schwenk, in New Directions in the Theory of Graphs, edited by F. Harary (Academic, New York, 1973), pp. 275–307] to conclude that ‘‘almost all trees are isospectral.’’