Search results for "math-ph"
showing 10 items of 525 documents
TIME-MINIMAL CONTROL OF DISSIPATIVE TWO-LEVEL QUANTUM SYSTEMS: THE INTEGRABLE CASE
2009
The objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two-level quantum systems whose dynamics is governed by the Lindblad equation. We focus our analysis on the case where the extremal Hamiltonian is integrable.
Partially implicit Runge-Kutta methods for wave-like equations
2012
In this work we present a new class of Runge-Kutta (RK) methods for solving systems of hyperbolic equations with a particular structure, generalization of a wave-equation. The new methods are {\it partially implicit} in the sense that a proper subset of the equations of the system contains some terms which are treated implicitly. These methods can be viewed as a particular case of the implicit-explicit (IMEX) RK methods for systems of equations with wave-like structure. For these systems, the optimal methods with the new structure are easier to derive than the IMEX ones, specially when aiming at higher-order (up to fourth-order in this work). The methods are constructed considering the clas…
Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function
2009
A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A s…
Structure of locally convex quasi C * -algebras
2008
There are examples of C*-algebras A that accept a locally convex *-topology τ coarser than the given one, such that Ã[τ] (the completion of A with respect to τ) is a GB*-algebra. The multiplication of A[τ] may be or not be jointly continuous. In the second case, Ã[*] may fail being a locally convex *-algebra, but it is a partial *-algebra. In both cases the structure and the representation theory of Ã[τ] are investigated. If Ã+ τ denotes the τ-closure of the positive cone A+ of the given C*-algebra A, then the property Ā+ τ ∩ (-Ā+ τ) = {0} is decisive for the existence of certain faithful *-representations of the corresponding *-algebra Ã[τ]
Solutions of the LPD equation and multi-parametric rogue waves
2022
Quasi-rational solutions to the Lakshmanan Porsezian Daniel equation are presented. We construct explicit expressions of these solutions for the first orders depending on real parameters. We study the patterns of these configurations in the (x, t) plane in function of the different parameters. We observe in the case of order 2, three rogue waves which move according to the two parameters. In the case of order 3, six rogue waves are observed with specific configurations moving according to the four parameters.
From particular polynomials to rational solutions to the mKdV equation
2022
Rational solutions to the modified Korteweg-de Vries (mKdV) equation are given in terms of a quotient of determinants involving certain particular polynomials. This gives a very efficient method to construct solutions. We construct very easily explicit expressions of these rational solutions for the first orders n = 1 until 10.
N order solutions with multi-parameters to the Boussinesq and KP equations and the degenerate rational case
2021
From elementary exponential functions which depend on several parameters, we construct multi-parametric solutions to the Boussinesq equation. When we perform a passage to the limit when one of these parameters goes to 0, we get rational solutions as a quotient of a polynomial of degree N (N + 1) − 2 in x and t, by a polynomial of degree N (N + 1) in x and t for each positive integer N depending on 3N parameters. We restrict ourself to give the explicit expressions of these rational solutions for N = 1 until N = 3 to shortened the paper. We easily deduce the corresponding explicit rational solutions to the Kadomtsev Petviashvili equation for the same orders from 1 to 3.
From particular polynomials to rational solutions to the PII equation
2022
The Painlevé equations were derived by Painlevé and Gambier in the years 1895 − 1910. Given a rational function R in w, w ′ and analytic in z, they searched what were the second order ordinary differential equations of the form w ′′ = R(z, w, w ′) with the properties that the singularities other than poles of any solution or this equation depend on the equation only and not of the constants of integration. They proved that there are fifty equations of this type, and the Painlevé II is one of these. Here, we construct solutions to the Painlevé II equation (PII) from particular polynomials. We obtain rational solutions written as a derivative with respect to the variable x of a logarithm of a…
Multi-parameters rational solutions to the mKdV equation
2021
N-order solutions to the modified Korteweg-de Vries (mKdV) equation are given in terms of a quotient of two wronskians of order N depending on 2N real parameters. When one of these parameters goes to 0, we succeed to get for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2N real parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 6.
Other patterns for the first and second order rational solutions to the KPI equation
2022
We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in x, y and t depending on several real parameters. We get an infinite hierarchy of rational solutions written as a quotient of a polynomial of degree 2N (N + 1) − 2 in x, y and t by a polynomial of degree 2N (N + 1) in x, y and t, depending on 2N − 2 real parameters for each positive integer N. We construct explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x, y) plane for different values of time t and parameters. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of…