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HexaKdV
[article]
2002
arXiv
pre-print
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An analog of the lattice KdV equation of Nijhoff et al. is constructed on a hexagonal lattice. The resulting system of difference equations exhibits soliton solutions with interesting local structure: there is a nontrivial phase shift on moving between adjacent lattice sites, with the magnitude of the shift tending to zero in the continuum limit.
arXiv:nlin/0209041v1
fatcat:fi2gfp64arbydiuk7aoiekihxa
Gaussian Quadrature without Orthogonal Polynomials
[article]
2005
arXiv
pre-print
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A novel development is given of the theory of Gaussian quadrature, not relying on the theory of orthogonal polynomials. A method is given for computing the nodes and weights that is manifestly independent of choice of basis in the space of polynomials. This method can be extended to compute nodes and weights for Gaussian quadrature on the unit circle and Gauss type quadrature rules with some fixed nodes.
arXiv:math/0506199v1
fatcat:g6xlrvtufff6dm23wd5wbiv4hq
Symmetries of KdV and Loop Groups
[article]
1996
arXiv
pre-print
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A simple version of the Segal-Wilson map from the SL(2,C) loop group to a class of solutions of the KdV hierarchy is given, clarifying certain aspects of this map. It is explained how the known symmetries, including Backlund transformations, of KdV arise from simple, field independent, actions on the loop group. A variety of issues in understanding the algebraic structure of Backlund transformations are thus resolved.
arXiv:solv-int/9606004v1
fatcat:jhm4kawxnbc7po5rbkjwxyniru
Commuting Extensions and Cubature Formulae
[article]
2004
arXiv
pre-print
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The specific matrices used can be found on the internet at http://www.math.biu.ac.il/∼schiff/commext.html. ...
The nodes and weights are available at http://www.math.biu.ac.il/∼schiff/commext.html.
Figure 2 2 displays the location of the nodes in the degree 15 and 17 formulae. ...
arXiv:math/0408076v1
fatcat:i5w5hmdv6jdf3oweq5nfag3jmq
Four Symmetries of the KdV equation
[article]
2021
arXiv
pre-print
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We identify 4 nonlocal symmetries of KdV depending on a parameter. We explain that since these are nonlocal symmetries, their commutator algebra is not uniquely determined, and we present three possibilities for the algebra. In the first version, 3 of the 4 symmetries commute; this shows that it is possible to add further (nonlocal) commuting flows to the standard KdV hierarchy. The second version of the commutator algebra is consistent with Laurent expansions of the symmetries, giving rise to
arXiv:2108.01550v1
fatcat:j7ys7lk6fvfybi734dc7uz4p6u
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... n infinite dimensional algebra of hidden symmetries of KdV. The third version is consistent with asymptotic expansions for large values of the parameter, giving rise to the standard commuting symmetries of KdV, the infinite hierarchy of "additional symmetries", and their traditionally accepted commutator algebra (though this also suffers from some ambiguity as the additional symmetries are nonlocal). We explain how the 3 symmetries that commute in the first version of the algebra can all be regarded as infinitesimal double B\"acklund transformations.
The Variance of Standard Option Returns
[article]
2012
arXiv
pre-print
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The vast majority of works on option pricing operate on the assumption of risk neutral valuation, and consequently focus on the expected value of option returns, and do not consider risk parameters, such as variance. We show that it is possible to give explicit formulae for the variance of European option returns (vanilla calls and puts, as well as barrier options), and that for American options the variance can be computed using a PDE approach, involving a modified Black-Scholes PDE. We show
arXiv:1204.3452v1
fatcat:zyc6yse4ubdudfc42eylyki7gm
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... w the need to consider risk parameters, such as the variance, and also the probability of expiring worthless (PEW), arises naturally for individual investors in options. Furthermore, we show that a volatility smile arises in a simple model of risk-seeking option pricing.
Actions for Integrable Systems and Deformed Conformal Theories
[article]
1992
arXiv
pre-print
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I report on work on a Lagrangian formulation for the simplest 1+1 dimensional integrable hierarchies. This formulation makes the relationship between conformal field theories and (quantized) 1+1 dimensional integrable hierarchies very clear.
arXiv:hep-th/9210137v1
fatcat:xs7yqhxo2rbnxojfeb6p6tz7ii
A Dynamical Systems Approach to The Fourth Painleve Equation
[article]
2018
arXiv
pre-print
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We use methods from dynamical systems to study the fourth Painleve equation PIV. Our starting point is the symmetric form of PIV, to which the Poincare compactification is applied. The motion on the sphere at infinity can be completely characterized. There are fourteen fixed points, which are classified into three different types. Generic orbits of the full system are curves from one of four asymptotically unstable points to one of four asymptotically stable points, with the set of allowed
arXiv:1810.09643v1
fatcat:ux5uhpqfdjeinjhu3oponucxde
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... itions depending on the values of the parameters. This allows us to give a qualitative description of a generic real solution of PIV.
Self-Dual Yang-Mills and the Hamiltonian Structures of Integrable Systems
[article]
1992
arXiv
pre-print
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In recent years it has been shown that many, and possibly all, integrable systems can be obtained by dimensional reduction of self-dual Yang-Mills. I show how the integrable systems obtained this way naturally inherit bihamiltonian structure. I also present a simple, gauge-invariant formulation of the self-dual Yang-Mills hierarchy proposed by several authors, and I discuss the notion of gauge equivalence of integrable systems that arises from the gauge invariance of the self-duality equations
arXiv:hep-th/9211070v1
fatcat:vpjva6svj5c2jbgbujruvozk4q
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... and their hierarchy); this notion of gauge equivalence may well be large enough to unify the many diverse existing notions.
The KdV Action and Deformed Minimal Models
[article]
1992
arXiv
pre-print
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An action is constructed that gives an arbitrary equation in the KdV or MKdV hierarchies as equation of motion; the second Hamiltonian structure of the KdV equation and the Hamiltonian structure of the MKdV equation appear as Poisson bracket structures derived from this action. Quantization of this theory can be carried out in two different schemes, to obtain either the quantum KdV theory of Kupershmidt and Mathieu or the quantum MKdV theory of Sasaki and Yamanaka. The latter is, for specific
arXiv:hep-th/9205105v2
fatcat:s6lwkzzd7rhxbojr7lmhowp6ny
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... lues of the coupling constant, related to a generalized deformation of the minimal models, and clarifies the relationship of integrable systems of KdV type and conformal field theories. As a generalization it is shown how to construct an action for the SL(3)-KdV (Boussinesq) hierarchy.
Commuting extensions and cubature formulae
2005
Numerische Mathematik
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Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A 1 , . . . , A d ,
doi:10.1007/s00211-005-0628-z
fatcat:hf334a34ijfuxhnuvq2fs3kree
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... elated to the coordinate operators x 1 , . . . , x d , in R d . We prove a correspondence between cubature formulae and "commuting extensions" of A 1 , . . . , A d , satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them.
Topological gauge theory and twistors
1989
Physics Letters B
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By use of harmonic superfields, Witten's topological gauge theory is written as a field theory on a supertwistor space. Further, using the realisation of complex compactified Minkowski space as a quadric in CP 5 , we show that the theory has a simple formulation in terms of analytic superfields on super-CP 5 × CP 3 .
doi:10.1016/0370-2693(89)91320-8
fatcat:7efwl2k4szcg3a3udb5qdsl7em
Zero curvature formulations of dual hierarchies
1996
Journal of Mathematical Physics
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Zero curvature formulations are given for the "dual hierarchies" of standard soliton equation hierarchies, recently introduced by Olver and Rosenau, including the physically interesting Fuchssteiner-Fokas-Camassa-Holm hierarchy.
doi:10.1063/1.531486
fatcat:z6xkbh6mjfda7mwzuv2wdw5fs4
On The Hamiltonian Structures and The Reductions of The KP Hierarchy
[article]
1992
arXiv
pre-print
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Recent work on a free field realization of the Hamiltonian structures of the classical KP hierarchy and of its flows is reviewed. It is shown that it corresponds to a reduction of KP to the NLS system. (Talk given by D.A.D. at the NSERC-CAP Workshop on Quantum Groups, Integrable Models and Statistical Systems, Kingston, Canada July 13-17 1992.)
arXiv:hep-th/9210080v1
fatcat:vqmsx4p4anccnaadk53cgrnqmi
Unified derivation of Bohmian methods and the incorporation of interference effects
[article]
2007
arXiv
pre-print
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We present a unified derivation of Bohmian methods that serves as a common starting point for the derivative propagation method (DPM), Bohmian mechanics with complex action (BOMCA) and the zero-velocity complex action method (ZEVCA). The unified derivation begins with the ansatz ψ=e^iS/ħ where the action, S, is taken to be complex and the quantum force is obtained by writing a hierarchy of equations of motion for the phase partial derivatives. We demonstrate how different choices of the
arXiv:0706.3508v1
fatcat:ode7vkikfbcwzhgs5wfh5hx37m
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... ry velocity field yield different formulations such as DPM, BOMCA and ZEVCA. The new derivation is used for two purposes. First, it serves as a common basis for comparing the role of the quantum force in the DPM and BOMCA formulations. Second, we use the new derivation to show that superposing the contributions of real, crossing trajectories yields a nodal pattern essentially identical to that of the exact quantum wavefunction. The latter result suggests a promising new approach to deal with the challenging problem of nodes in Bohmian mechanics.
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