Journal of Mathematical Physics
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First order analytic difference equations and integrable quantum systems We present a new solution method for a class of first order analytic difference equations. The method yields explicit “minimal” solutions that are essentially unique. Special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic, trigonometric and elliptic type—Euler’s gamma function being of rational type. We study these generalized gamma functions in considerable detail. The scattering and weight functions (u- and w-functions) associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions. We obtain detailed information on these u- and w-functions, exploiting the difference equations they satisfy.
Journal of Mathematical Physics - Tập 38 Số 2 - Trang 1069-1146 - 1997
Inverse scattering for lasso graph The inverse problem for the magnetic Schrödinger operator on the lasso graph with different matching conditions at the vertex is investigated. It is proven that the Titchmarsh-Weyl function known for different values of the magnetic flux through the cycle determines the unique potential on the loop, provided the entries of the vertex scattering matrix S parametrizing matching conditions satisfy s12s23s31 ≠ s13s21s32. This is in contrast to numerous examples showing that the potential on the loop cannot be reconstructed from the boundary measurements.
Journal of Mathematical Physics - Tập 54 Số 4 - 2013
Two-dimensional lumps in nonlinear dispersive systems Two-dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev–Petviashvili and a two-dimensional nonlinear Schrödinger type equation. The amplitude of these solutions is rational in its independent variables. These solutions are constructed by taking a ’’long wave’’ limit of the corresponding N-soliton solutions obtained by direct methods. The solutions describing multiple collisions of lumps are also presented.
Journal of Mathematical Physics - Tập 20 Số 7 - Trang 1496-1503 - 1979
Complexiton solutions to soliton equations by the Hirota method We apply the Hirota direct method to construct complexiton solutions (complexitons). The key is to use Hirota bilinear forms. We prove that taking pairs of conjugate wave variables in the 2N-soliton solutions generates N-complexion solutions. The general theory is used to construct multi-complexion solutions to the Korteweg–de Vries equation.
Journal of Mathematical Physics - Tập 58 Số 10 - 2017
Solving linear stochastic differential equations The aim of this paper is to provide the user with tools for the solution of linear differential equations with random coefficients. Only analytic methods which lead to expressions in closed form for first and second order moments and probability distributions of the solution are considered. The paper deals both with approximate methods which require the existence of a small (or large) dimensionless parameter and with the method of model coefficients, where the true coefficients of the stochastic equation are replaced by random step functions with the same first and second order moments and probability distributions, chosen in such a way that the equation can be solved analytically. The second procedure does not rely on the existence of a small parameter.
Journal of Mathematical Physics - Tập 15 Số 5 - Trang 524-534 - 1974
An exact solution to Einstein’s equations with a stiff equation of state A solution to the equations of general relativity is given which is spherically-symmetric and nonstatic with an inhomogeneous density profile ρ and a pressure p given by the stiff equation of state p=ρc2. The solution may be of use in representing collapsed astrophysical systems or the early stages of an inhomogeneous cosmology.
Journal of Mathematical Physics - Tập 19 Số 11 - Trang 2283-2284 - 1978
Random walk and the heat equation on superspace and anyspace Random walks are used to study diffusion on anyspace. Anyspace is characterized by coordinate ξ with ξN=0 and statistics ξξ′=e2πi/Nξ′ξ between independent copies. Anyonic integration and anyonic Dirac δ functions are introduced, and reduced to familiar results for supersymmetry when N=2. These ingredients are then used to formulate and solve the resulting anyonic diffusion equation.
Journal of Mathematical Physics - Tập 35 Số 7 - Trang 3753-3760 - 1994
AKNS hierarchy, Darboux transformation and conservation laws of the 1D nonautonomous nonlinear Schrödinger equations By constructing nonisospectral Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy, we investigate the nonautonomous nonlinear Schrödinger (NLS) equations which have been used to describe the Feshbach resonance management in matter-wave solitons in Bose–Einstein condensate and the dispersion and nonlinearity managements for optical solitons. It is found that these equations are some special cases of a new integrable model of nonlocal nonautonomous NLS equations. Based on the Lax pairs, the Darboux transformation and conservation laws are explored. It is shown that the local external potentials would break down the classical infinite number of conservation laws. The result indicates that the integrability of the nonautonomous NLS systems may be nontrivial in comparison to the conventional concept of integrability in the canonical case.
Journal of Mathematical Physics - Tập 52 Số 4 - 2011
Schrödinger Scattering Amplitude. I The Schrödinger scattering amplitude for a fixed potential is studied as a function of the three components of the initial momentum, the three components of the final momentum, and the square root of the energy.
Journal of Mathematical Physics - Tập 2 Số 5 - Trang 710-713 - 1961
Analytic Continuation of the Off-Energy-Shell Scattering Amplitude It is shown that the two-body scattering amplitude 〈k| T(E) |k′〉 may be analytically continued in E through the physical cut into a region of meromorphy, provided that the potential satisfies certain requirements. The residue at a pole is a separable operator whose form agrees with previous work. The proof also demonstrates the existence of a region of meromorphy of 〈k| T(E) |k′〉 as a function of seven complex variables.
Journal of Mathematical Physics - Tập 8 Số 4 - Trang 873-877 - 1967
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