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We investigate the time evolution of Hagedorn wavepackets by non-Hermitian quadratic Hamiltonians. We state a direct connection between coherent states and Lagrangian frames. For the time evolution, a multivariate polynomial recursion is derived that describes the activation of lower lying excited states, a phenomenon unprecedented for Hermitian propagation. Finally we apply the propagation of excited states to the Davies–Swanson oscillator. [ABSTRACT FROM AUTHOR]

Multiparametric families of hypergeometric τ-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at t = 0 shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is generated that is adapted to the Grassmannian element associated with the τ-function. Multicurrent correlators are defined in terms of the τ-function and shown to provide an alternative generating function for weighted Hurwitz numbers. Fermionic vacuum state expectation value representations are provided for the adapted bases, pair correlators, and multicurrent correlators. Choosing the weight generating function as a polynomial and restricting the number of nonzero “second” KP flow parameters in the Toda τ-function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite “window” of adapted basis elements. The pair correlator is shown to provide a Christoffel-Darboux type finite rank integrable kernel, and the WKB series coefficients of the associated adjoint system are computed recursively, leading to topological recursion relations for the generators of the weighted Hurwitz numbers. [ABSTRACT FROM AUTHOR]

Based on solving the Lenard recursion equation and the zero-curvature equation, we derive the modified Sawada-Kotera (SK) hierarchy associated with a 3 × 3 matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix for the modified SK hierarchy, we introduce a trigonal curve Km-1 and present the corresponding Baker-Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the modified SK flows. With the aid of the property of the Baker-Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. Algebro-geometric solutions of the entire modified SK hierarchy are obtained by using asymptotic expansion of the meromorphic function and its Riemann theta function representation. As an application, we give two simple examples. [ABSTRACT FROM AUTHOR]

A new harmonic oscillatory ring-shaped noncentral potential is proposed in which the noncentral electric dipole and a novel angle-dependent component are included. The pseudospin symmetry for this potential is investigated by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator. The resulting three-term recursion relation for the expansion coefficients of the wavefunctions (both angular and radial) is presented. The angular/radial wavefunction is written in terms of the Jacobi/Laguerre polynomials. The discrete spectrum of the bound states is obtained by diagonalization of the radial recursion relation. [ABSTRACT FROM AUTHOR]

The present work studies structural properties of the recursion operator of the noncommutative KdV equation. As the main result, it is proved that this operator is hereditary. The notion of hereditary operators was introduced by Fuchssteiner for infinite-dimensional integrable systems, building on classical concepts from differential topology. As an illustration for the consequences of this property, it is deduced that the flows of the noncommutative KdV hierarchy mutually commute. [ABSTRACT FROM AUTHOR]

The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold ['Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods,' J. Math. Phys. 50, 073510 (2009)]. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies. [ABSTRACT FROM AUTHOR]

We study matrix models in the β-ensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first β-deformed corrections in the one-cut and the two-cut cases, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in N=1 gauge theories, and the calculation of vevs of surface operators in superconformal N=2 theories and their Liouville duals. Finally, we study the β-deformation of the Chern-Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Ω-deformed topological string on the resolved conifold, and therefore that the β-deformation might provide a different generalization of topological string theory in toric Calabi-Yau backgrounds. [ABSTRACT FROM AUTHOR]

Conventional nondegenerate perturbation theory for some nth state starts with the corresponding unperturbed state. The present formulation yields recursively perturbation expansions for any bound state using the sole information of the unperturbed ground state. Logarithmic perturbation theory is exploited along with supersymmetric quantum mechanics to achieve this end. As the method involves ground-state perturbations of a series of supersymmetric Hamiltonians, concern about nodal shifts of targeted excited states arises only at the ultimate step, thus, minimizing considerably the labor of clumsy computations involved in dealing with excited states. [ABSTRACT FROM AUTHOR]

In the present paper we consider a new two-component fifth-order integrable system recently found by Mikhailov, Novikov, and Wang, and show that this system possesses a hereditary recursion operator and infinitely many commuting symmetries and conservation laws, as well as infinitely many compatible Hamiltonian and symplectic structures, and is therefore completely integrable. The system in question admits a reduction to the Kaup-Kupershmidt equation. [ABSTRACT FROM AUTHOR]

The Schrödinger equation with noncentral electric dipole ring-shaped potential is investigated by working in a complete square integrable basis that supports an infinite tridiagonal matrix representation of the wave operator. The three-term recursion relations for the expansion coefficients of both the angular and radial wavefunctions are presented. The discrete spectrum for the bound states is obtained by the diagonalization of the radial recursion relation. Some potential applications of this system in different fields are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the binary Bell polynomials are applied to succinctly construct bilinear formulism, bilinear Bäcklund transformations, Lax pairs, and Darboux covariant Lax pairs for the (2+1)-dimensional breaking soliton equation. An extra auxiliary variable is introduced to get the bilinear formulism. The infinitely local conservation laws of the equation are found by virtue of its Lax equation and a generalized Miura transformation. All conserved densities and fluxes are given with explicit recursion formulas. [ABSTRACT FROM AUTHOR]

In this paper, a new integrable two-component system, mt=[m(uxvx-uv+uvx-uxv)]x,nt=[n(uxvx - uv + uvx - uxv)]x, where m = u - uxx and n=v-vxx, is proposed. Our system is a generalized version of the integrable system mt=[m(ux2-u2)]x, which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained. [ABSTRACT FROM AUTHOR]

The supersymmetric analog of the reciprocal transformation is introduced. This is used to establish a transformation between one of the supersymmetric Harry Dym equations and the supersymmetric modified Korteweg-de Vries equation. The reciprocal transformation, as a Bäcklund-type transformation between these two equations, is adopted to construct a recursion operator for the supersymmetric Harry Dym equation. By proper factorization of the recursion operator, a bi-Hamiltonian structure is found for the supersymmetric Harry Dym equation. Furthermore, a supersymmetric Kawamoto equation is proposed and is associated with the supersymmetric Sawada-Kotera equation. The recursion operator and odd bi-Hamiltonian structure of the supersymmetric Kawamoto equation are also constructed. [ABSTRACT FROM AUTHOR]

Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection. A key tool is the application of Bäcklund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1+1-dimensions are considered, a “Bäcklund chart” depicts links relating, in particular, the Korteweg–de Vries (KdV) to the modified KdV (mKdV) hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered. Then Bäcklund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework. [ABSTRACT FROM AUTHOR]

We present a new symplectic structure and a hereditary recursion operator for the Sasa-Satsuma equation which is widely used in nonlinear optics. Using an integrodifferential substitution relating this equation to a third-order symmetry flow of the complex sine-Gordon II equation enabled us to find a hereditary recursion operator and higher Hamiltonian structures for the latter equation. We also show that both the Sasa-Satsuma equation and the third-order symmetry flow for the complex sine-Gordon II equation are bi-Hamiltonian systems, and we construct several hierarchies of local and nonlocal symmetries for these systems. [ABSTRACT FROM AUTHOR]

We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number. [ABSTRACT FROM AUTHOR]

We present further properties of a previously proposed recursive scheme for parametrization of n-by-n unitary matrices. We show that the factors in the recursive formula may be introduced in any desired order. The method is used to study the invariant phases of unitary matrices. The case of four-by-four unitary matrices is investigated in detail. We also address the question of how to construct symmetric unitary matrices (i.e., unitary matrices U that satisfy the condition Uij=Uji) using the recursive approach. [ABSTRACT FROM AUTHOR]

A simple recursive scheme for parametrization of n-by-n unitary matrices is presented. The n-by-n matrix is expressed as a product containing the (n-1)-by-(n-1) matrix and a unitary matrix that contains the additional parameters needed to go from n-1 to n. The procedure is repeated to obtain recursion formulas for n-by-n unitary matrices. [ABSTRACT FROM AUTHOR]

The integrability of a coupled KdV-mKdV system is tested by means of singularity analysis. The true Lax pair associated with this system is obtained by the use of prolongation technique. [ABSTRACT FROM AUTHOR]

A systematic investigation on the construction of recursion operators for partial differential-difference equations (PDDEs) with two independent variables (one continuous and one discrete) using its generalized symmetries is presented. Also it is explained how to factorize the obtained recursion operators. The applicability of the above procedure have been illustrated for the relativistic toda (RT), BelovChaltikian (BC) and Blaszak-Marciniak (BM) lattice equations and shown that the former two lattice equations admit (2 × 2) matrix recursion operators while the latter one possesses a (3 × 3) matrix recursion operator. Furthermore, the constructed recursion operators can be written as a factor of 2 distinct invertible matrix operators in each of the lattice equations. It is also proved explicitly that the factorized operators are Hamiltonian and hence RT, BC and BM lattice equations are bi-Hamiltonian systems. [ABSTRACT FROM AUTHOR]

We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a nonstandard Lax representation. We give several examples for N = 2 and N = 3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators. We also discuss a reduction of N + 1 systems to N systems of some new equations of hydrodynamic type. [ABSTRACT FROM AUTHOR]

Comments are made on the meaning and existence of recursion operators for completely integrable systems. It is shown that any nonresonant integrable system admits infinitely many Hamiltonian descriptions and recursion operators. [ABSTRACT FROM AUTHOR]

On the basis of the Schwinger–Dyson variational method the recursion relations of the character expansion coefficients of the SU(N) gauge action are derived. Three recursion relations that are associated with the Casimir invariants of the second, third, and fourth orders are shown to be enough to determine the character expansion coefficient of every representation recursively in SU(N). [ABSTRACT FROM AUTHOR]

In this article, an identity satisfied by the so-called recursion operator is derived. The identity generates by itself an infinite sequence of Lax pairs, thus ensuring the complete integrability of the corresponding hierarchy of nonlinear evolution equations. It is also shown that this identity yields the familiar property that the squares of eigenfunctions of the associated linear spectral problem satisfy the linearized version of the respective soliton equation. [ABSTRACT FROM AUTHOR]

On the basis of the Schwinger–Dyson variational method the recursion relations of the character expansion coefficients of the U(N) gauge action are derived. Three recursion relations that are associated with the Casimir invariants are shown to be enough to determine the coefficients completely. [ABSTRACT FROM AUTHOR]

The Lie algebra of symmetries of the Landau–Lifshitz equation (LL) is discussed. Point, higher-order, and nonlocal symmetries of the LL are computed. The recursion operator is constructed from a nonlocal symmetry, reflecting the bi-Hamiltonian structure of the LL. The same recursion operator of the LL has been derived by Barouch, Fokas, and Papageorgiou [J. Math. Phys. 29, 2628 (1988)], starting from the Lax pair. [ABSTRACT FROM AUTHOR]

Olver’s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator. Some negative order KdV equations are derived in this way. In particular, the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated Schrödinger equation used in the inverse scattering solution of the KdV equation and this formulation is used to show that this new equation is the Miura transform of a sinh-Gordon equation. [ABSTRACT FROM AUTHOR]

The strict mathematical derivation is given for the recursion relations of the coefficients of the character expansion of the single-link action in the SU(2), U(2), and SU(3) groups. This relies on the immediate Schwinger–Dyson variation on the character. It will be suggested that the independent recursion relations for the U(N) and SU(N) groups are associated with the group invariants, i.e., the Casimir invariants and the unit generator [for U(N)]. [ABSTRACT FROM AUTHOR]

A general scheme of fractal decomposition of exponential operators is presented in any order m. Namely, exp[x(A+B)]=Sm(x)+O(xm+1) for any positive integer m, where Sm(x)=et1A et2B et3A et4B···etMA with finite M depending on m. A general recursive scheme of construction of {tj} is given explicitly. It is proven that some of {tj} should be negative for m≥3 and for any finite M (nonexistence theorem of positive decomposition). General systematic decomposition criterions based on a new type of time-ordering are also formulated. The decomposition exp[x(A+B)]=[Sm(x/n)]n +O(xm+1/nm) yields a new efficient approach to quantum Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

When r/a≊0, Löwdin’s α function (1/r)αl(fLM|a,r) is expressed as (1/r)αl ( fLM|a,r) =2γ(LM|l )aL-1∑∞n=l [∑imaxi=l+M[hn,2n-i (LM|l )/(2n-i)!]a2n-ig(2n-i) (a)](r/a)2n-l, where a is a separation between a new center placed on the origin and the old center located at a given point on the z axis that is the origin of the coordinate system defining a function f (R)YML(Θ,[uppercase_phi_synonym]) to be expanded around the new center [ f(R) is an arbitrary radial function, and YML(Θ,[uppercase_phi_synonym]), a complex spherical harmonics], and r is a distance from the new center. Here, imax=min{2n,2(L+l )}, and the function g(j)(a) is expressed as g(j)(a)=[(d/dR)j ( f(iR)/RL-1)]R=a.The closed form of the coefficients hn,2n-i(LM|l ) is given by Eq. (27) in J. Math. Phys. 26, 3193 (1985) (referred to as Part II). Since they are expressed in terms of the different coefficients bKk(LM|l ) independent of the form of f(R), hn,2n-i(LM|l ) constitute a set of universal constants as do bKk(LM|l ). The explicit expression for the factor γ(LM|l ) in front of the summation symbol is given by Eq. (2.9) in J. Math. Phys. 25, 1133 (1984). Hereafter the coefficients expressed by hn,2n-i(LM|l )/(2n-i)! are symbolized by In,2n-i(LM|l ). Although they are expressed in a complex form with a triple sum, that expression can be reduced to two considerably simple forms by changing the summation indices to others by using the addition theorem for binomial coefficients and the condition for the sum in this theorem to vanish. Moreover, in the two special cases that i=l+M and i=2(L+l ) or 2(L+l )-1, those two expressions are reduced to two single-term forms, respectively.Introducing into the expression for In,2n-i(LM|l ) in terms of bKk(LM|l ) the recursion formula for bKk(LM|l ) in only M given by Eq. (24) in Part II, leads to the recursion formula for In,2n-i(LM|l ) with respect to M. On the other side, the... [ABSTRACT FROM AUTHOR]

Some U(n) Racah coefficients appearing in various applications of vector coherent state and K-matrix theories are calculated: For such a purpose, use is made of their definition in terms of U(n):U(n-1) reduced Wigner coefficients and U(n-1) Racah coefficients. By starting from known U(2) Racah coefficients, the recursion relations obtained are solved by induction over n. [ABSTRACT FROM AUTHOR]

The recursion operator and the bi-Hamiltonian structure for an integrable two-dimensional version of the Toda chain are algorithmically derived from the corresponding linear problem. The intimate relation between two-dimensional theories and non-Abelian one-dimensional theories is emphasized. The analogies and differences between discrete and continuous integrable two-dimensional systems are pointed out and discussed. [ABSTRACT FROM AUTHOR]

A hierarchy of local nonlinear evolution equations associated with a new spectral problem is derived. It is shown that each equation is Hamiltonian and that their fluxes commute and a local infinite set of conserved densities is given. An interesting reduction is considered. In this case a hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse. Also this hierarchy satisfies a canonical geometrical scheme. It contains as a special case the Pohlmeyer–Lund–Regge equation. [ABSTRACT FROM AUTHOR]

Higher-order or generalized symmetries for the super modified KdV equation are constructed. Moreover, by the introduction of graded potentials nonlocal symmetries are obtained, one of them leading to the recursion operator for symmetries in a straightforward way. [ABSTRACT FROM AUTHOR]

For a class of Hamiltonian systems there exist infinite series of non-Hamiltonian symmetries. Some properties of these series are illustrated using a Boussinesq equation. It is shown that the recursion operators generated by these non-Hamiltonian symmetries are powers of the original recursion operator. A class of recursion formulas for the constants of the motion (not for the corresponding symmetries!) is given. [ABSTRACT FROM AUTHOR]