Efficient numerical method for solving the direct Zakharov–Shabat scattering problem (original) (raw)
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An example of full solution of the inverse scattering problem on the half line is presented. For this purpose, a simple analytically solvable model system (Morse potential) is used, which is expected to be a reasonable approximation to a real potential. First one calculates all spectral characteristics for the fixed model system. This way one gets all the necessary input data (otherwise unobtainable) to implement powerful methods of the inverse scattering theory. In this paper, the multi-step procedure to solve the Marchenko integral equation is described in full details. Excellent performance of the method is demonstrated and its combination with the Marchenko differential equation is discussed. In addition to the main results, several important analytic properties of the Morse potential are unveiled. For example, a simple analytic algorithm to calculate the phase shift is derived.
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The numerical discretization of the Zakharov-Shabat Scattering problem using integrators based on the implicit Euler method, trapezoidal rule and the split-Magnus method yield discrete systems that qualify as Ablowitz-Ladik systems. These discrete systems are important on account of their layer-peeling property which facilitates the differential approach of inverse scattering. In this paper, we study the Darboux transformation at the discrete level by following a recipe that closely resembles the Darboux transformation in the continuous case. The viability of this transformation for the computation of multisoliton potentials is investigated and it is found that irrespective of the order of convergence of the underlying discrete framework, the numerical scheme thus obtained is of first order with respect to the step size.
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The Marchenko method is developed in the inverse scattering problem for a linear system of first-order differential equations containing potentials proportional to the spectral parameter. The corresponding Marchenko system of integral equations is derived in such a way that the method can be applied to some other linear systems for which a Marchenko method is not yet available. It is shown how the potentials and the scattering solutions to the linear system are constructed from the solution to the Marchenko system. The bound-state information for the linear system with any number of bound states and any multiplicities is described in terms of a pair of constant matrix triplets. When the potentials in the linear system are reflectionless, some explicit solution formulas are presented in closed form for the potentials and for the scattering solutions to the linear system. The theory is illustrated with some explicit examples.
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A fourth-order multi-exponential scheme is proposed for the Zakharov-Shabat system. The scheme represents a product of 13 exponential operators. The construction of the scheme is based on a fourth-order three-exponential scheme, which contains only one exponent with a spectral parameter. This exponent is factorized to the fourth-order with the Suzuki formula of 11 exponents. The obtained scheme allows the use of a fast algorithm in calculating the initial problem for a large number of spectral parameters and conserves the quadratic invariant exactly for real spectral parameters.
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We modify the J-matrix technique for scattering so that problems with long-range interactions are easily solved. This is done by introducing additional terms in the asymptotic three-term recurrence relation that take into account asymptotic effects of the potential. The solutions of this modified recurrence relation are a very good approximation of the exact scattering solution. Only a small number of residual coefficients need to be calculated. As a result, the numerical effort to solve the scattering problem is seriously reduced. The technique is illustrated with a Yukawa potential.
Rapid solution of integral equations of scattering theory in two dimensions
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The present paper describes an algorithm for rapid solution of boundary value problems for the Helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory. CPU time requirements of previously published algorithms of this type are of the order FZ', where n is the number of nodes in the discretization of the boundary of the scatterer. The CPU time requirements of the algorithm of the present paper are #3, and can be further reduced, making it considerably more practical for large scale problems.