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The Poincare equation, a second-order partial differential equation describing wave motions in a rotating spheroid of arbitrary eccentricity satisfying a certain set of the boundary condition, is studied. A new polynomial as the general solution of the Poincare equation in spheroidal geometry is found for the first time. The paper focuses on some unusual and intriguing mathematical properties of the new Poincare polynomial. The possible completeness of the set of eigenfunctions of the Poincare equation in the form of the new polynomial is also discussed. The new Poincare polynomial would provide a powerful basis for the mathematical analysis in many important geophysical and astrophysical problems.

The Γ-convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions. We determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability. In two dimensions a threshold exists for the stability of the spot solution. It is stable if the sample size is small and unstable if the sample size is large. The stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems. In one study no two-interface ring solutions are found by the Γ-convergence method if the sample is small. A stable two-interface ring solution exists if the sample size is increased. It becomes unstable if the sample size is increased further.

We develop an operator method which helps finding exact solutions to nonlinear evolution equations (NLEs). Our working schema goes as follows: First we translate the given (NLE) into an appropriate operator version (ONLE). Second, we look for solutions to (ONLE) of the form U=(I+L)−1M, where both L and M are operator-valued functions of the space–time variables and the range of M locates in some appropriate Banach algebras which admits a functional φ that preserves the squares [i.e., φ(A2)=φ(A)2]. Finally, a solution u of the given (NLE) can be obtained by setting u≔φ(U). This method is named by the LM method. Using the LM method, we have rederived the famous Cole–Hopf transformation which reduces the nonlinear Burgers equation into the linear heat equation. The main part of this article is to use the LM method to study the vector Korteweg–de Vries (KdV) equations ut=uxxx+3(u2)x settled in finite-dimensional unital Banach algebras J. It is shown that these vector KdV equations admit soliton solutions. Spe...

A possible means to classify the interaction between the Weyl and Ricci tensors is to look at the number of principal null directions that the Weyl and Plebanski tensors have in common. This paper presents algebraic conditions that can be used to determine this number without explicitly calculating the principal null directions themselves.

Liouville integrable noncanonical Hamiltonian systems with variable coefficient symplectic forms are generated from the AKNS 3×3 matrix Lax pairs by binary symmetry constraints of the AKNS hierarchy. These integrable systems provide new integrable factorization for every AKNS system in the hierarchy.

The asymptotics (in the width of windows) of eigenvalues and bands for two-dimensional waveguides and three-dimensional layers coupled through small windows is obtained. The technique is matching of asymptotic expansions of the solutions of boundary value problems.

The so$(2,1)$ Lie algebra is applied to three classes of two- and three-dimensional Smorodinsky-Winternitz super-integrable potentials for which the path integral discussion has been recently presented in the literature. We have constructed the Green's functions for two important super-integrable potentials in $R^{2}.$ Among the super-integrable potentials in $R^{3}$, we have considered two examples, one is maximally super-integrable and another one minimally super-integrable. The discussion is made in various coordinate systems. The energy spectrum and the suitably normalized wave functions of bound and continuous states are then deduced., Comment: 35 pages, no figures

The properties of linear equations on braided linear spaces are investigated and conservation laws for them are derived. The conserved currents are given in the explicit form. The procedure is then applied to scalar wave equations on a quantum plane and on q-Minkowski space.

In the present paper, we prove that the family of exponentials associated to the eigenvalues of the perturbed operator T(ɛ) ≔ T0 + ɛT1 + ɛ2T2 + … + ɛkTk + … forms a Riesz basis in L2(0, T), T > 0, where ɛ∈C, T0 is a closed densely defined linear operator on a separable Hilbert space H with domain D(T0) having isolated eigenvalues with multiplicity one, while T1, T2, … are linear operators on H having the same domain D⊃D(T0) and satisfying a specific growing inequality. After that, we generalize this result using a H-Lipschitz function. As application, we consider a non-selfadjoint problem deduced from a perturbation method for sound radiation.

We study the Jacobi unitary ensemble defined by the perturbed Chebyshev weight w(x)=(1−x2)−12(tn2−x2)α, x ∈ (−1, 1), tn > 1. The algebraic singularity coalesces with the end point of the support of the weight function, namely, tn → 1, the hard edge, as n → ∞. We use the Riemann-Hilbert approach (the method of Deift and Zhou) to obtain the asymptotics of the polynomials orthogonal with respect to w(x). We show that the universality property is preserved: the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel. The main result is on the local behavior at the edge of the spectrum. The limit kernel at the edge, when tn − 1 = O(n−2), is described by the ψ −function for a specific solution of the Painleve V equation. We also show that when tn varies to a fixed t > 1, the Painleve V kernel transits to the Bessel kernel J−12. On the other hand, when tn approaches 1, the Painleve V kernel can be approximated by another Bessel kernel Jα−12.

We investigate spectral properties of spherical Schrodinger operators (also known as Bessel operators) with δ-point interactions concentrated on a discrete set. We obtain necessary and sufficient conditions for these Hamiltonians to be self-adjoint, lower-semibounded and also we investigate their spectra. We also extend the classical Bargmann estimate to such Hamiltonians. In certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities. We apply our results to Schrodinger operators in L2(Rn) with a singular interaction supported by an infinite family of concentric spheres.

Coupled electromagnetic TE and TM wave propagation in a nonlinear plane layer is considered. Nonlinearity inside the layer is described by Kerr law. Physical problem is reduced to a nonlinear two-parameter eigenvalue problem for a system of (nonlinear) ordinary differential equations. It is proved that TE and TM waves that form (nonlinear) coupled TE-TM wave can propagate at different frequencies ωE and ωM, respectively. These frequencies can be chosen independently. Existence of coupled surface TE and TM waves is proved. Intervals of localization of coupled eigenvalues are found.

The purpose of this article is two-fold. First, we obtain some certain bounds for the sums of (positive and negative) powers of the eigenvalues of the clamped plate problem of the Dirichlet bi-Laplacian operator Δ2|D, restricted to a bounded domain D⊂Rd with d ⩾ 2. Second, we establish lower bounds for the sums of eigenvalues of Δ2|D sharper than the bounds recently obtained by Cheng and Wei [“A lower bound for eigenvalues of a clamped plate problem,” Calculus Var. Partial Differ. Equ. 42(3–4), 579–590 (2011)]10.1007/s00526-011-0399-6. All these estimates are sharp in the sense of Weyl asymptotics.

Explicit flows associated to any tangent vector fields on any isospectral manifold for the AKNS operator acting in L2 × L2 on the unit interval are written down. The manifolds are of infinite dimension (and infinite codimension). The flows are called isospectral and also are Hamiltonian flows. It is proven that they may be explicitly expressed in terms of regularized determinants of infinite matrix-valued functions with entries depending only on the spectral data at the starting point of the flow. The tangent vector fields are decomposed as ∑ξkTk where ξ ∈ l2 and the Tk ∈ L2 × L2 form a particular basis of the tangent vector spaces of the infinite dimensional manifold. The paper here is a continuation of Amour [“Explicit isospectral flows for the AKNS operator on the unit interval,” Inverse Probl. 25, 095008 (2009)]10.1088/0266-5611/25/9/095008 where, except for a finite number, all the components of the sequence ξ are zero in order to obtain an explicit expression for the isospectral flows. The regulariz...

The procedure for finding the solutions of the Vakhnenko-Parkes equation by means of the inverse scattering method is described. The continuous spectrum is taken into account in the associated eigenvalue problem. The suggested special form of the singularity function for continuous part of the spectral data gives rise to the multi-mode solutions. The sufficient conditions are proved in order that these solutions become real functions. The interaction of the N periodic waves is studied. The procedure is illustrated by considering a number of examples.

Some years ago, Mikhailov and Sokolov identified as integrable the neat system of two evolution equations U=V2, V=U2, where U ≡ U(t) and V ≡ V(t) are two N × N matrices, N is an arbitrary positive integer, t (“time”) is the independent variable, and superimposed dots indicate the time derivatives. This entails, rather trivially, that the generic solution of the modified version of this model reading U=V2+iωU, V=U2+iωV, with ω an arbitrary positive constant, is completely periodic with period T = 2π/ω (or possibly a period which is an integer multiple of T): “isochrony.” Another, less trivial, consequence of their finding is the observation that the solution of the many-body problem characterized by the Hamiltonian system of N Newtonian evolution equations, xn=−a2xn5+g22∑m=1,m≠nN[(xn−xm)−3+xn+xm−3],n=1,...,N, where xn ≡ xn(t) are N scalar dependent variables and a, g are two arbitrary constants, is simply related to the evolution of the (appropriately rescaled) eigenvalues of a matrix simply related t...

We derive a dispersion equation for determining eigenvalues of inhomogeneous quantum wells in terms of spectral parameter power series and apply it for the numerical treatment of eigenvalue problems. The method is algorithmically simple and can be easily implemented using available routines of such environments for scientific computing as MATLAB.

By studying tensor powers ϕ⊗n of a positive map ϕ of B(K) into B(H) for finite dimensional Hilbert spaces K and H, we give conditions for a map ϕ, such that ϕ⊗n 2-positive for all n to be nondistillable and completely positive.

Using non-skew-symmetric classical r-matrices r12(u,v) with spectral parameters, we generalize Fuchsian systems and Schlesinger equations (isomonodromy equations) viewing the last as the nonautonomous Hamiltonian equations corresponding to the generalized Gaudin Hamiltonians. Quantizing these systems we construct generalization of Knizhnik–Zamolodchikov equations corresponding to the non-skew-symmetric classical r-matrices r12(u,v) with spectral parameters. In the case of an ordinary skew-symmetric classical r-matrix r12(u−v) depending on the difference of the spectral parameters, we reobtain, as special partial cases, a standard generalization of Fuchsian systems, Schlesinger, and Knizhnik–Zamolodchikov equations.

A bijective correspondence is constructed between the parabolic vector bundles with rational parabolic weights on a Riemann surface X and the holomorphic vector bundles on quasiregular Sasakian threefolds over X. This bijective correspondence preserves the stable objects.

The consistency condition associated with the feasibility of setting out the twistor equation in curved space-times is considered within the frameworks of the Infeld–van der Waerden γe-formalisms. It appears that the e-formalism version of the condition bears a simpler form because of the applicability of a peculiar differential property of spin-tensor densities whose weights and antiweights are suitably related to their valences. Certain calculational techniques are likewise utilized for deriving a set of new wave equations.

We write the spherical curl transformation for Trkalian fields using differential forms. Then we consider Radon transform of these fields. The Radon transform of a Trkalian field satisfies a corresponding eigenvalue equation on a sphere in transform space. The field can be reconstructed using knowledge of the Radon transform on a canonical hemisphere. We consider relation of the Radon transformation with Biot-Savart integral operator and discuss its transform introducing Radon-Biot- Savart operator. The Radon transform of a Trkalian field is an eigenvector of this operator. We also present an Ampere law type relation for these fields. We apply these to Lundquist solution. We present a Chandrasekhar-Kendall type solution of the corresponding equation in the transform space. Lastly, we focus on the Euclidean topologically massive Abelian gauge theory. The Radon transform of an anti-self-dual field is related by antipodal map on this sphere to the transform of the self-dual field obtained by inverting space coordinates. The Lundquist solution provides an example of quantization of topological mass in this context., Comment: 23 pages

The Darboux integrability method is particularly useful to determine first integrals of nonplanar autonomous systems of ordinary differential equations, whose associated vector fields are polynomials. In particular, we obtain first integrals for a variant of the generalized Raychaudhuri equation, which has appeared in string inspired modern cosmology.

In this paper, we consider the blow up phenomenon of the nonclassical diffusion equation. Under suitable conditions on the initial data, we prove a finite time blow up result.

We derive a new method of calculating the mean negative moments and ratios of squared modulus of characteristic polynomials of a certain class of random matrices. New results obtained with this method are presented.

In this paper I investigate the existence of nontrivial breather solutions of the discrete nonlinear Schrodinger equation with the unbounded potential at infinity. First I derive a discrete version of compact embedding theorem. Then combining the Nehari manifold approach and the compact embedding theorem, I show the existence of breather solutions without Palais–Smale condition. The results on the exponential decay of breather solutions are also provided in this paper.

We consider the eigenvalues of the non-self-adjoint Zakharov-Shabat systems as the coupling constant of the potential is varied. In particular, we are interested in eigenvalue collisions and eigenvalue trajectories in the complex plane. We identify shape features in the potential that are responsible for the occurrence of collisions and we prove asymptotic formulas for large coupling constants that tell us where eigenvalues collide or where they emerge from the continuous spectrum. Some examples are provided which show that the asymptotic methods yield results that compare well with exact numerical computations.

A method for determining the eigenfunctions of the equation ∇×B=λB for finite cylindrical geometry with normal boundary condition B∙n=0 and nonaxisymmetric modes ∼eimθ,m≠0 was given by Morse [J. Math. Phys. 46, 113511 (2005)] This method uses series solution of a Chandrasekhar-Kendall [Astrophys. J. 126, 157 (1957)] function. Here this method is generalized to include annular (coaxial) cylindrical geometry of finite length. The method is also applied to the case of rectangular duct geometry with periodic z-dependence. For slender aspect ratios in the annular cylinder, the eigenfunctions and eigenfields correspond to the rectangular duct. The eigenfunctions for finite-length rectangular boxes can also be obtained from the rectangular duct case. The robust convergence properties (an∼1∕n4) and simple eigenvalue equations from Morse are retained in these other geometries.

This paper is concerned with localization properties of coherent states. Instead of classical uncertainty relations we consider "generalized" localization quantities. This is done by introducing measures on the reproducing kernel. In this context we may prove the existence of optimally localized states. Moreover, we provide a numerical scheme for deriving them.

Dynamical systems on Rn presenting geometric chaos, i.e., open domains where bounded and unbounded orbits are intermingled, have been constructed. The opposite situation (open scattering) has been studied for integrable Hamiltonian and non-Hamiltonian vector fields and for equations of type x=−∇V(x) when the potential has the form V=a(r)+b(r)F(θ) in spherical coordinates.

We study a pair of neutrally stable eigenvalues of zero energy in the linearized NLS equation. We prove that the pair of isolated eigenvalues, where each eigenvalue has geometric multiplicity one and algebraic multiplicity N, is associated with 2P negative eigenvalues of the energy operator, where P=N∕2 if N is even and P=(N−1)∕2 or P=(N+1)∕2 if N is odd. When the potential of the linearized NLS problem is perturbed due to parameter continuations, we compute the exact number of unstable eigenvalues that bifurcate from the neutrally stable eigenvalues of zero energy.

The necessary and sufficient conditions are derived for the reducibility of a hydrodynamic type system into a block-diagonal form with k mutually interacting blocks. Applications to the perturbations of the Benney system and to the Hamiltonian systems of partial differential equations are presented. The algebraic identities connecting the Nijenhuis tensors NB(A)(u,v) and NA(u,v) and Haantjes tensors HB(A)(u,v) and HA(u,v) are discovered.

In the present work we use the method of equivalence to determine necessary and sufficient conditions for a general fourth order ordinary differential equation to be equivalent to the flat model under a fiber preserving transformation. As a result, explicit and simple conditions are obtained.

In this paper, a general method of treating Hamiltonians of deformed nanoscale systems is proposed. This method is used to derive a second-order approximation both for the strong and weak formulations of the eigenvalue problem. The weak formulation is needed in order to allow deformations that have discontinuous first derivatives at interfaces between different materials. It is shown that, as long as the deformation is twice differentiable away from interfaces, the weak formulation is equivalent to the strong formulation with appropriate interface boundary conditions. It is also shown that, because the Jacobian of the deformation appears in the weak formulation, the approximations of the weak formulation is not equivalent to the approximations of the strong formulation with interface boundary conditions. The method is applied to two one-dimensional examples (a sinusoidal and a quantum-well potential) and one two-dimensional example (a freestanding quantum wire), where it is shown that the energy eigenvalu...

This paper is part of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in real or complex conformally flat spaces. Here we consider classical superintegrable systems with nondegenerate potentials in three dimensions. We show that there exists a standard structure for such systems, based on the algebra of 3×3 symmetric matrices, and that the quadratic algebra always closes at order 6. We show that the spaces of truly second-, third-, fourth-, and sixth-order constants of the motion are of dimension 6, 4, 21, and 56, respectively, and we construct explicit bases for the fourth- and sixth order constants in terms of products of the second order constants.

In this note, we explicitly compute the functional determinant of a Dirac Laplacian with nonlocal pseudodifferential boundary conditions over a finite cylinder in terms of the ζ-function of the Dirac operator on the cross section and the pseudodifferential operators defining the boundary conditions. In particular, this result reduces to our previous formula [J. Phys. AJPHAC5 37, 7381 (2004)] for the special case of generalized Atiyah–Patodi–Singer conditions. To prove our main result, we use the gluing and comparison formulas established by the present authors in Refs 14 and 15.

The canonical transformation approach generated by the semisimple subgroup GCM(3)⊂Sp(6,R) is applied to reduction of the Lie–Poisson bracket on coadjoint orbits of the Sp(6,R) group and the Poisson coalgebra spO*(6) is determined. Investigating the construction of the N-particle phase, induced by this reduction, we identify the Poisson coalgebra spO*(6) as the algebra of quadratic O(K), K≡N−1 invariant forms on symplectic 6 K−12 dimensional phase space T*[O(K−3)\O(K)], K⩾3. The general classification scheme of Poisson orbits for spO*(6) is found and applied to the classification of coadjoint orbits of the Sp(6,R) group occurring in the decomposition of N-particle phase spaces. We show that the spO*(k), k=4,6 Poisson action on some class of surfaces determined by Casimir invariants is not transitive. The Poisson maps for all classes of orbits spO*(4) and spO*(6) are found. The quantum unitary irreducible representations of spO*(4) are obtained.

In this paper, we use renormalization methods to study self-similarity in the fluctuations ηi of the Harper equation in the strong-coupling limit for quadratic irrationals of the form (a2+4−a)/2 for a∈N. Using the decimation method, we obtain a second-order functional recurrence which we prove rigorously has an entire fixed point. This fixed point governs the scaling of the fluctuations ηi in the strong-coupling limit.

We decompose the Hilbert space of wave functions into two subspaces, and assign to a given observable two effective representatives that act in the model space. The first serves to determine some of the eigenvalues of the full observable, while the second serves to determine its matrix elements, in any basis in one of the subspaces, in terms of quantities pertaining to the model space. We also show that if the Hamiltonian of a physical system possesses symmetries then these symmetries continue to hold for its effective representatives of the first type. Maximum information about the system can be obtained in terms of two sets of effective representatives. The first set of representatives is complete. Other observables that do not commute with all members of the complete set have only one type of representative.

In view of the geometrical importance of spaces with constant scalar curvature, a systematic study of spherically symmetric such space–time manifolds with respect to the eigenvalues of the Ricci tensor is made. The cases of two double or one quadruple eigenvalue are treated exhaustively. In the generic case of one double and two single eigenvalues, no conformally flat solutions, and only solutions with one arbitrary function of one variable are found. We also give all four-dimensional decomposable s.s. spaces with constant curvature scalar.