Integration of a generalized H\'enon-Heiles Hamiltonian

The generalized H\'enon-Heiles Hamiltonian $H=1/2(P_X^2+P_Y^2+c_1X^2+c_2Y^2)+aXY^2-bX^3/3$ with an additional nonpolynomial term $\mu Y^{-2}$ is known to be Liouville integrable for three sets of values of $(b/a,c_1,c_2)$. It has been previously integrated by genus two theta functions only in one of these cases. Defining the separating variables of the Hamilton-Jacobi equations, we succeed here, in the two other cases, to integrate the equations of motion with hyperelliptic functions.Comment: LaTex 2e. To appear, Journal of Mathematical Physic

On the gravitational field of static and stationary axial symmetric bodies with multi-polar structure

We give a physical interpretation to the multi-polar Erez-Rozen-Quevedo solution of the Einstein Equations in terms of bars. We find that each multi-pole correspond to the Newtonian potential of a bar with linear density proportional to a Legendre Polynomial. We use this fact to find an integral representation of the $\gamma$ function. These integral representations are used in the context of the inverse scattering method to find solutions associated to one or more rotating bodies each one with their own multi-polar structure.Comment: To be published in Classical and Quantum Gravit

Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles

The probabilities for gaps in the eigenvalue spectrum of the finite dimension $N \times N$ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $N \to \infty$, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2

Factorization of Ising correlations C(M,N) for ν= −k \nu= \, -k and M+N odd, M≤NM \le N, T<TcT < T_c and their lambda extensions

We study the factorizations of Ising low-temperature correlations C(M,N) for $\nu=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $M \neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlev\'e VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlev\'e VI equation. For both the two and four factor case, we find that there is a one parameter family of boundary conditions on the Okamoto sigma-form of Painlev\'e VI equations which generalizes the factorization of the correlations C(M,N) to an additive decomposition of the corresponding sigma's solutions of the Okamoto sigma-form of Painlev\'e VI equation which we call lambda extensions. At a special value of the parameter, the lambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials in the complete elliptic functions of the first and second kind. We also generalize some Tracy-Widom (Painlev\'e V) relations between the sum and difference of sigma's to this Painlev\'e VI framework.Comment: 45 page

Late Cretaceous to Recent Deformation Related to Inherited Structures and Subsequent Compression within the Persian Gulf: A 2D Seismic Case Study

The Persian Gulf is part of an asymmetric foreland basin related to the Zagros Orogen. Few published studies of this basin and associated onshore areas include seismic reflection data. We present a seismic-stratigraphic interpretation based on marine 2D seismic data, which reveals the presence of two types of compressional structures within the basin: (1) faulted domes related to salt movement and the offshore trace of a NNE–SSW-trending dextral basement fault (the Kazerun Fault); (2) long-wavelength (16 km), low-amplitude (60 ms two-way travel time) folds relating to the advancing deformation front associated with the orogen. Thinning of age-constrained stratal units across structures related to the offshore trace of the Kazerun Fault implies a distinct pulse of uplift on this fault during the Maastrichtian. The geometry of growth strata across other intra-basin structures suggests a second, later stage of deformation, which began in the Middle Miocene. Thickening and folding of post-Middle Miocene stratal units towards the NE (i.e. towards the Zagros Orogen) is interpreted to reflect rapid loading, subsidence and compression related to southwestwards advance of the orogen. The results of this study have implications for the interaction between pre-existing structures and later compressional events both within the Persian Gulf and elsewhere

Late Cretaceous to Recent Deformation Related to Inherited Structures and Subsequent Compression within the Persian Gulf: A 2D Seismic Case Study

The Persian Gulf is part of an asymmetric foreland basin related to the Zagros Orogen. Few published studies of this basin and associated onshore areas include seismic reflection data. We present a seismic-stratigraphic interpretation based on marine 2D seismic data, which reveals the presence of two types of compressional structures within the basin: (1) faulted domes related to salt movement and the offshore trace of a NNE–SSW-trending dextral basement fault (the Kazerun Fault); (2) long-wavelength (16 km), low-amplitude (60 ms two-way travel time) folds relating to the advancing deformation front associated with the orogen. Thinning of age-constrained stratal units across structures related to the offshore trace of the Kazerun Fault implies a distinct pulse of uplift on this fault during the Maastrichtian. The geometry of growth strata across other intra-basin structures suggests a second, later stage of deformation, which began in the Middle Miocene. Thickening and folding of post-Middle Miocene stratal units towards the NE (i.e. towards the Zagros Orogen) is interpreted to reflect rapid loading, subsidence and compression related to southwestwards advance of the orogen. The results of this study have implications for the interaction between pre-existing structures and later compressional events both within the Persian Gulf and elsewhere

Constructing Integrable Third Order Systems:The Gambier Approach

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlev\'e method for ODE's and the singularity confinement method for mappings).Comment: 14 pages, TEX FIL

Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients

We consider deformations of $2\times2$ and $3\times3$ matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which don't satisfy the well-known system of Schlesinger equations (or its natural generalization). Some general statements concerning reducibility of such deformations for $2\times2$ ODEs are proved. An explicit example of the general non-Schlesinger deformation of $2\times2$-matrix ODE of the Fuchsian type with 4 singular points is constructed and application of such deformations to the construction of special solutions of the corresponding Schlesinger systems is discussed. Some examples of isomonodromy and non-isomonodromy deformations of $3\times3$ matrix ODEs are considered. The latter arise as the compatibility conditions with linear ODEs with non-singlevalued coefficients.Comment: 15 pages, to appear in J. Phys.