Hyperbolic Differential Operators Research Articles

On Distribution of Rice–Middleton Model

The probability density function in Rice–Middleton model, which describes the behavior of the single sinusoidal random signal combined with Gaussian noise is expressed in three mutually independent ways: firstly, with the aid of an integral representation of the modified Bessel function of the first kind of integer order; secondly, by a hyperbolic cosine differential operator and thirdly, applying the Grünwald–Letnikov fractional derivative. The cumulative distribution functions are also described in all these cases, and also using the Nuttall Q–function. An associated, seemingly new, probability distribution is introduced which cumulative distribution function and the raw moments of general real order are obtained whilst the characteristic function’s power series form is inferred. The exposition ends with a discussion in which by–product summations are given for the considered Neumann series of the second type built by modified Bessel functions of the second kind having integer order.

Boundedness of planar jump discontinuities for homogeneous hyperbolic systems

Suppose that [Formula: see text] is a homogeneous constant coefficient strongly hyperbolic partial differential operator on [Formula: see text] and that [Formula: see text] is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of [Formula: see text], the characteristic variety of [Formula: see text] is the graph of a real analytic function [Formula: see text] with [Formula: see text] identically equal to zero or the maximal possible value [Formula: see text]. Suppose that the source function [Formula: see text] is compactly supported in [Formula: see text] and piecewise smooth with singularities only on [Formula: see text]. Then the solution of [Formula: see text] with [Formula: see text] for [Formula: see text] is uniformly bounded on [Formula: see text]. Typically when [Formula: see text] on the conormal variety, the sup norm of the jump in the gradient of [Formula: see text] across [Formula: see text] grows linearly with [Formula: see text].

Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds

Let $\Gamma$ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$, and $\square$ the Laplacian which is a second-order hyperbolic differential operator. We study linear independence of a family of generalized Poincaré series introduced by Kassel-Kobayashi [Adv. Math. 287 (2016), 123-236, arXiv:1209.4075], which are defined by the $\Gamma$-average of certain eigenfunctions on $\mathrm{AdS}^{3}$. We prove that the multiplicities of $L^{2}$-eigenvalues of the hyperbolic Laplacian $\square$ on $\Gamma\backslash\mathrm{AdS}^{3}$ are unbounded when $\Gamma$ is finitely generated. Moreover, we prove that the multiplicities of stable $L^{2}$-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.

On the resolvent existence and the separability of a hyperbolic operator with fast growing coefficients in L2(R2)

This paper studies the question of the resolvent existence, as well as, the smoothness of elements from the definition domain (separability) of a class of hyperbolic differential operators defined in an unbounded domain with greatly increasing coefficients after a closure in the space L2(R2). Such a problem was previously put forward by I.M. Gelfand for elliptic operators. Here, we note that a detailed analysis shows that when studying the spectral properties of differential operators specified in an unbounded domain, the behavior of the coefficients at infinity plays an important role.

Factorization of the Main Hyperbolic Differential Operator of the Micropolar Elasticity Theory

The article deals with the dynamic equations of the linear micropolar theory of elasticity. They are related with respect to vector fields of displacements and microrotations. Coupled equations are also obtained for vortex vector potentials of displacements and microrotations. Using (or not) calibration conditions, it is possible to formulate uncoupled equations that include the same main differential operator. Conditions for the hyperbolicity of the main operator and a condition connecting the defining constants that ensures its factorizability, that is, representation as a product of simpler differential operators commuting with each other. The actual construction of the operator factors in the factorized form is carried out.

Exact controllability for the semilinear Mindlin-Timoshenko system

We consider the dynamical one-dimensional Mindlin–Timoshenko system for beams. We obtain a global exact controllability result for this semilinear system with superlinear nonlinearities. For this purpose, we establish an observability estimate for the linearized system with bounded potentials. Moreover, we obtain an explicit estimate of the observability constant in terms of the norms of the potentials. Such an estimate is obtained by a combination of a pointwise estimate, a global Carleman estimate for hyperbolic differential operators and an analysis of the regularity of solutions of a auxiliary optimal control problem. The controllability of the semilinear system is proved using the Hilbert Uniqueness Method (HUM) and a fixed point technique.

Maxwell’s equations, the Euler index, and Morse theory

We show that the singularities of the Fresnel surface for Maxwell’s equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to desingularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles.

Invariants of hyperbolic partial differential operators

We present a construction of a large class of Laplace invariants for linear hyperbolic partial differential operators of fairly general form and arbitrary order.

The microlocal spectrum condition, initial value formulations, and background independence

We analyze implications of the microlocal spectrum/Hadamard condition for states in a (linear) quantum field theory on a globally hyperbolic spacetime M in the context of a (distributional) initial value formulation. More specifically, we work in 3+1-split M ≅ ℝ × Σ and give a bound, independent of the spacetime metric, on the wave front sets of the initial data for a quasi-free Hadamard state in a quantum field theory defined by a normally hyperbolic differential operator P acting in a vector bundle E→πM. This aims at a possible way to apply the concept of Hadamard states within approaches to quantum field theory/gravity relying on a Hamiltonian formulation, potentially without a (classical) background metric g.

Linear hyperbolic PDEs with noncommutative time

Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form $(D+\lambda W)f=0$ are studied, where $D$ is a normal or prenormal hyperbolic differential operator on ${\mathbb R}^n$, $\lambda\in\mathbb C$ is a coupling constant, and $W$ is a regular integral operator with compactly supported kernel. In particular, $W$ can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small $|\lambda|$, the hyperbolic character of $D$ is essentially preserved. Unique advanced/retarded fundamental solutions are constructed by means of a convergent expansion in $\lambda$, and the solution spaces are analyzed. It is shown that the acausal behavior of the solutions is well-controlled, but the Cauchy problem is ill-posed in general. Nonetheless, a scattering operator can be calculated which describes the effect of $W$ on the space of solutions of $D$. It is also described how these structures occur in the context of noncommutative Minkowski space, and how the results obtained here can be used for the analysis of classical and quantum field theories on such spaces.

A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators

The behavior of orbits of the Hamilton vector field H p of the principal symbol p of a second-order hyperbolic differential operator is discussed. In our previous paper, assuming that p is noneffectively hyperbolic on the doubly characteristic manifold Σ of p , we have proved that if H S 3 p = 0 on Σ with the Hamilton vector field H S of some specified S , then there exists a bicharacteristic landing on Σ tangentially. The aim of this paper is to provide a much more simple proof of this result since the previous proof was fairly long and rather complicated.

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

Abstract The non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

A system of coupled partial differential equations exhibiting both elevation and depression rogue wave modes

Abstract Analytical solutions are obtained for a coupled system of partial differential equations involving hyperbolic differential operators. Oscillatory states are calculated by the Hirota bilinear transformation. Algebraically localized modes are derived by taking a Taylor expansion. Physically these equations will model the dynamics of water waves, where the dependent variable (typically the displacement of the free surface) can exhibit a sudden deviation from an otherwise tranquil background. Such modes are termed ‘rogue waves’ and are associated with ‘extreme and rare events in physics’. Furthermore, elevations, depressions and ‘four-petal’ rogue waves can all be obtained by modifying the input parameters.

ON THE CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS: THE GEVREY 3 WELL-POSEDNESS

For hyperbolic differential operators P with double characteristics we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and the Hamilton map and flow of the associated principal symbol p. If the Hamilton map admits a Jordan block of size 4 on the double characteristic manifold denoted by Σ and by assuming that the Hamilton flow does not approach Σ tangentially, we proved earlier that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 4 for any lower order term. In the present paper, we remove this restriction on the Hamilton flow and establish that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 3 for any lower order term and we check that the Gevrey index 3 is optimal. Combining this with results already proved for the other cases, we conclude that the Hamilton map and flow completely characterizes the threshold for the strong Gevrey well-posedness and vice versa.

A criterion for the nonexistence of closed leaves in unorientable planar foliations

The family of all smooth foliations F on an open set Ω ⊂ R 2 ≅ C is naturally parameterized by all smooth maps X : Ω → S 1 = { z ∈ C : | z | = 1 } , in the sense that the values ± X ( p ) determine the tangent line to the leaf of F at p ∈ Ω . If F is further assumed to be orientable, a smooth global branch Y of the square root of X can be chosen. In this case, one has the classical Lyapunov criterion: if there is a real-valued u ∈ C 1 ( Ω ) such that Y u = Re { 2 Y ¯ u z ¯ } is nowhere zero, then F has no closed leaves (the vector field Y has no periodic orbits). In this paper we introduce an analytic criterion for the nonexistence of closed leaves, similar in spirit to that of Lyapunov, but which allows for F to be unorientable as well. The possible lack of orientability makes the replacement for the first order differential operator Y considerably more involved. In fact, one has to work with a second order linear hyperbolic differential operator L F whose coefficients carry information about the curvature of the leaves of F . It is shown that if F is given by X : Ω → S 1 in the manner described above, and there exists a real-valued u ∈ C 2 ( Ω ) such that L F u : = Im { 2 X ¯ u z ¯ z ¯ + ( X z ¯ − X ¯ X z ) u z ¯ } is nowhere zero, then F has no closed leaves. We apply the new criterion when X is holomorphic, providing also an example that shows the need for the first order term in the definition of L F .

On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness

For a hyperbolic second-order differential operator P, we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and some algebraic and geometric properties of the principal symbol p. If the Hamilton map Fp of p (the linearization of the Hamilton field Hp along double characteristics) has nonzero real eigenvalues at every double characteristic (the so-called effectively hyperbolic case), then it is well known that the Cauchy problem for P is well posed in any Gevrey class 1≤s<+∞ for any lower-order term. In this paper we prove that if p is noneffectively hyperbolic and, moreover, such that KerFp2∩ImFp2≠{0} on a C∞ double characteristic manifold Σ of codimension 3, assuming that there is no null bicharacteristic landing Σ tangentially, then the Cauchy problem for P is well posed in the Gevrey class 1≤s<4 for any lower-order term (strong Gevrey well-posedness with threshold 4), extending in particular via energy estimates a previous result of Hörmander in a model case.

New realization of the pseudoconvexity and its application to an inverse problem

We consider a hyperbolic differential operator with variable principal term. We first give a sufficient condition for the pseudoconvexity which yields a Carleman estimate and a necessary condition. Our sufficient condition implies that level sets generated by the weight function in the Carleman estimate are convex with respect to the set of rays given by a 0(x), and give a more general explicit condition of a 0 for the pseudoconvexity. Second we apply the Carleman estimate to an inverse problem of determining a 0 by Cauchy data on a lateral boundary with relaxed constraints on a 0.

The Deformation Quantizations of the Hyperbolic Plane

We describe the space of (all) invariant, both formal and non-formal, deformation quantizations on the hyperbolic plane $${\mathbb D}$$ as solutions of the evolution of a second order hyperbolic differential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a unified method producing every quantization of $${\mathbb D}$$ , and provides, in the 2-dimensional context, an exact solution to Weinstein’s WKB quantization program within geometric terms. The construction reveals the existence of a metric of Lorentz signature canonically attached (or ‘dual’) to the geometry of the hyperbolic plane through the quantization process.

Exact Controllability for Multidimensional Semilinear Hyperbolic Equations

In this paper, we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients. For this purpose, we establish an observability estimate for the linear hyperbolic equation with an unbounded potential, in which the crucial observability constant is estimated explicitly by a function of the norm of the potential. Such an estimate is obtained by a combination of a pointwise estimate and a global Carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem.

On well-posedness of the mixed problem for nonstrictly hyperbolic operators

By using a new concept of polynomial pencil, this paper obtains new sufficient conditions for the well-posedness of the Cauchy problem for a nonstrictly hyperbolic differential operator.