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101.
A method to linearize the initial value problem of the Painlevé equations IV, V is given. The procedure involves formulating a Riemann-Hilbert boundary value problem on intersecting lines for the inverse monodromy problem. This boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space. Schlesinger transformations allow one to completely cover the parameter space. Special solutions are constructed from special cases of the Riemann problem as well. 相似文献
102.
We study the cohomology of the Schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra. We prove that, for all pseudodifferential operators in 3+1 dimensions of this type, the Schwinger term is equivalent to the twisted Radul cocycle, a modified version of the Radul cocycle arising in non-commutative differential geometry. In the process we also show how the ordinary Radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component. 相似文献
103.
104.
For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one
prescribes an initial condition plus either one boundary condition if q
t
and q
xxx
have the same sign (KdVI) or two boundary conditions if q
t
and q
xxx
have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing
the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q
x
(0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q
x
(0,t),q
xx
(0,t)} and {q
xx
(0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary
values, as well as with the function Φ
(t)(t,k), where Φ
(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation
for Φ
(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation
for Φ
(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function
Φ
(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications
of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation. 相似文献
105.
A.G. Sifalakis A.S. Fokas S.R. Fulton Y.G. Saridakis 《Journal of Computational and Applied Mathematics》2008
A new approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was introduced in Fokas [A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 53 (1997) 1411–1443]. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet to Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. This is based on the analysis of the so-called global relation, an equation which couples known and unknown components of the derivative on the boundary and which is valid for all values of a complex parameter k. A collocation-type numerical method for solving the global relation for the Laplace equation in an arbitrary bounded convex polygon was introduced in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. Here, by choosing a different set of the “collocation points” (values for k), we present a significant improvement of the results in Fulton et al. [An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004) 465–483]. The new collocation points lead to well-conditioned collocation methods. Their combination with sine basis functions leads to a collocation matrix whose diagonal blocks are point diagonal matrices yielding efficient implementation of iterative methods; numerical experimentation suggests quadratic convergence. The choice of Chebyshev basis functions leads to higher order convergence, which for regular polygons appear to be exponential. 相似文献
106.
With the use of the equations of motion of massless fields moving in a curved Friedmann-Robertson-Walker universe, we show, in some simple cases, that the energy-momentum tensor of a maximally 3-space symmetric distribution of the fields (i.e., an incoherent averaging over a complete set of modes of the field propagating in a Robertson-Walker background) has the standard perfect fluid form. As far as we know such an explicit demonstration, as well as the establishment of the compatibility of the equations of motion of the gravitational field with such an incoherently averaged source in the standard cosmology, has not previously been presented in the literature. Our results are found to hold for any value of the spatial curvature of the universe. 相似文献
107.
Numerical calculations of (finite temperature) superfluid vortex ring propagation against a particulate sheet show that the solid particle trajectories collapse to a very good approximation to the normal-fluid path lines. We propose an experiment in which, by measuring the solid particles' velocities, direct information about the instantaneous normal-fluid velocity values could be obtained. 相似文献
108.
A series of diverse indole-based chemotypes were synthesized from -tetrahydrocarboline (-THC) scaffolds prepared from commercially and readily available tryptamines and -ketoesters. Diversity can be generated within these chemotypes through the following strategies: (a) appendage of substituents to the -THC scaffold, prepared in situ or as a template, through further elaboration and (b) skeletal modifications to the -THC scaffold via ring forming or ring breaking reactions. The strategies described here are amenable to high throughput solution-phase parallel synthesis, providing access to novel indole-based screening libraries for drug discovery.Dedicated to Professor Spyros P. Perlepes 相似文献
109.
We investigate numerically the Navier-Stokes dynamics of reconnecting vortex rings at small Reynolds number for a variety of configurations. We find that reconnections are dissipative due to the smoothing of vorticity gradients at reconnection kinks and to the formation of secondary structures of stretched antiparallel vorticity which transfer kinetic energy to small scales where it is subsequently dissipated efficiently. In addition, the relaxation of the reconnection kinks excites Kelvin waves which due to strong damping are of low wave number and affect directly only large scale properties of the flow. 相似文献
110.
This paper uses Lie-Bäcklund operators to study the connection between classical and quantum-mechanical invariants and their relations to symmetry and to separation of variables. The problem of an isomorphic correspondence between classical and quantum mechanics is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl's transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl's transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered. 相似文献