摘要:

We investigate what happens if in the Feynman‐Kac functional, we perform the time integration with respect to a Borel measureηrather than ordinary Lebesgue measurel. Letu(t) be the operator associated with this functional through path integration. We show thatu(t), considered as a function of timet, satisfies a certain Volterra‐Stieltjes integral equation. This result establishes a “FeynmanKac formula with Lebesgue‐Stieltjes measureη.” One recovers the classical Feynman‐Kac formula by lettingη=l. We deduce from the integral equation thatu(t) satisfies a differential equation associated with the continuous partμofηwhenη=μ=l, this differential equation reduces to the heat or the Schrödinger equation in the probabilistic or quantum‐mechanical case, respectively. Moreover, we observe a new phenomenon, due to the discrete partvofη: the functionu(t) undergoes a discontinuity at every point in the support ofv, assumed here to be finite. Further, one obtains an explicit expression foru(t) in terms of operators alternatively associated withμandv. Our results are new even in the probabilistic or “imaginary time” case and allow us to unify various concepts. The derivation of our integral equation has an interesting combinatorial structure and makes essential use of the “generalized Dyson series”— recently introduced by G. W. Johnson and the author—that “disentangle” the operatoru(t). We provide natural physical interpretations of our results in both the diffusion and quantum‐mechanical cases. We also suggest further connections with FeynmanȈs opera

ISSN:0022-2526    

DOI:10.1002/sapm198776293

年代:1987

数据来源: WILEY

摘要:

A considerable amount of information is currently available on the creation and propagation of large solitary waves in marine straits. In order to be able to analyze such data we develop a theoretical model, extending previous one‐dimensional models to the case of straits with varying width and depth, and nonvanishing vorticity. Starting from the Euler equations for a three‐dimensional homogeneous incompressible inviscid fluid, we derive, in the quasi‐one‐dimensional long‐wave and shallow‐water approximation, a generalized KadomtsevPetviashvili (GKP) equation, together with its appropriate boundary conditions. In general, the coefficients of this equation depend on the form of the bottom and on the vorticity; the sides of the straits figure only in the boundary conditions. Under certain restrictions on the vorticity and the geometry of the straits we reduce the GKP equation to one of several completely integrable partial differential equations, in order to study the evolution of solitons which originate in

ISSN:0022-2526    

DOI:10.1002/sapm1987762133

年代:1987

数据来源: WILEY

摘要:

Finite‐amplitude wave propagation is considered in flows of boundary‐layer type when the wavelength is long compared to the boundary layer thickness. In this limit, the evolution of the amplitude is governed by the Benjamin‐Ono equation and we have computed the coefficients of its nonlinear and dispersive terms for the specific case of Tietjens's model. The propagation of wave packets is also considered, and it is found that for packets centered about anO(1) wavenumber questions again arise relative to long waves, except that now the packet‐induced mean flow is the “long wave.” By introducing an appropriate scaling for the far field and employing multiple scales in the direction transverse to the flow, it is shown how the mean‐flow distortion can be made to vanis

ISSN:0022-2526    

DOI:10.1002/sapm1987762169

年代:1987

数据来源: WILEY