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Matrix Riemann-Hilbert Problems and Maxwell-Bloch Equations without Spectral Broadening
Abstract
The Maxwell-Bloch equations have been intensively studied by many authors. The main results are based on the inverse scattering transform and the Marchenko integral equations. However this method is not acceptable for mixed problems. In the paper, we develop a method allowing to linearize mixed problems. It is based on simultaneous spectral analysis of both Lax equations and the matrix Riemann-Hilbert problems. We consider the case of infinitely narrow spectral line, i.e., without spectrum broadening. The proposed matrix Riemann-Hilbert problem can be used for studying temporal/spatial asymptotics of the solutions of Maxwell-Bloch equations by using a nonlinear method of steepest descent.
... If one chooses the sign "minus", then the problem is considered in a stable medium, in the so-called attenuator (for example, a model of self-induced transparency). The matrix Riemann-Hilbert problems were studied for this case in [18,20]. If the sign "plus" is chosen, then the problem is considered in an unstable medium (for example, a model of a two-level laser amplifier), which is the subject of our study. ...
... Formula (20) and Lemmas 3, 5 imply the following properties of Z(t, x, λ) = (Z[1](t, x, λ) Z [2](t, x, λ)): ...
A mixed initial-boundary value problem for nonlinear Maxwell-Bloch (MB) equations without spectral broadening is studied by using the inverse scattering transform in the form of the matrix Riemann-Hilbert (RH) problem. We use transformation operators whose existence is closely related with the Goursat problems with nontrivial characteristics. We also use a gauge transformation which allows us to obtain Goursat problems of the canonical type with rectilinear characteristics, the solvability of which is known. The transformation operators and a gauge transformation are used to obtain the Jost type solutions of the Ablowitz-Kaup-Newel-Segur equations with well-controlled asymptotic behavior by the spectral parameter near singular points. A well posed regular matrix RH problem in the sense of the feasibility of the Schwartz symmetry principle is obtained. The matrix RH problem generates the solution of the mixed problem for MB equations. © M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko), 2017.
The modified Korteweg-de Vries equation on the line is considered. The initial function is a discontinuous and piecewise constant step function, i.e. q(x,0)=c r for x≥0 and q(x,0)=c l for x<0, where c l , c r are real numbers which satisfy c l >c r >0. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t→∞. Using the steepest descent method we deform the original oscillatory matrix Riemann-Hilbert problem to explicitly solvable model forms and show that the solution of the initial-value problem has different asymptotic behavior in different regions of the xt plane. In the regions x<-6c l 2 t+12c r 2 t and x>4c l 2 t+2c r 2 t the main term of asymptotics of the solution is equal to c l and c r , respectively. In the region (-6c l 2 +12c r 2 )t<x<(4c l 2 +2c r 2 )t the asymptotics of the solution takes the form of a modulated hyper-elliptic wave generated by an algebraic curve of genus 2.
We consider the sine-Gordon equation in laboratory coordinates with both x and t in [0,∞). We assume that u(x,0), u t (x,0), u(0,t) are given, and that they satisfy u(x,0)→2πq, u t (x,0)→0, for large x, u(0,t)→2πp for large t, where q,p are integers. We also assume that u x (x,0), u t (x,0), u t (0,t), u(0,t)-2πp, u(x,0)-2πq∈L 2 . We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large t shows how the boundary conditions can generate solitons.
We consider the nonlinear Schrödinger equation in the variable q(x,t) with both x and t in [0,∞). We assume that q(x,0)=u(x) and q(0,t)=v(t) are given, that u(0)=v(0), and that u(x) and v(t) as well as their first two derivatives belong to L 1 ∩L 2 (ℝ + ). We show that the solution of this initial-boundary value problem can be reduced to solving a Riemann-Hilbert (RH) problem in the complex k-plane with jumps on Im(k 2 )=0. This RH problem is equivalent to a linear integral equation which has a unique global solution. This linear integral equation is uniquely defined in terms of certain functions (scattering data) b(k) and c(k). The function b(k) can be effectively computed in terms of u(x). However, although the analytic properties of c(k) are completely determined, the relationship between c(k), u(x) and v(t) is highly nonlinear. In spite of this difficulty, we can give an effective description of the asymptotic behavior of q(x,t) for large t. In particular, we show that as t→∞, solitons are generated moving away from the boundary. In addition, our formalism can be used to generate effectively pairs of functions q(0,t) and q x (0,t) compatible with a given q(x,0) as well as to determine the associated q(x,t). It is important to emphasize that the analysis of this problem, in addition to techniques of exact integrability, requires the essential use of general partial differential equations techniques.
It is shown that the stipulation of lossless propagation provides
sufficient asymptotic information on the pulse-propagation process to
permit determination of both amplitude and phase of a class of coherent
optical pulses (2nπ pulses, n=0, 1, 2, ...) by the inverse method.
Results for n=1 and 2 are summarized.
A model for optical pulse propagation in a two-level medium having relaxation times which are long compared to pulse length is summarized. Pulse shapes are described by solutions of a single nonlinear partial differential equation. Particular solutions are obtained by employing a Baecklund transformation.
A formal method of solving the initial---boundary-value problem in the quadrant x>=0, t>=0 for the Maxwell-Bloch system of equations is presented in terms of the inverse scattering problem.
A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann-Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine-Gordon equation in light-cone coordinates, the nonlinear Schrodinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrodinger equation is derived.
In this article we present a new and general approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, when evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg-de Vries (MKdV) equation