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We develop direct and inverse scattering theory for one-dimensional Schroedinger operators with steplike potentials which are asymptotically close to different finite-gap periodic potentials on different half-axes.
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A. Cohen and T. Kappeler, Scattering and inverse scattering for steplike potentials in the Schrödinger equation
1985
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R.G. Newton, Inverse scattering by a local impurity in a periodic potential in one dimension, II
1985
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N.E. Firsova, A direct and inverse scattering problem for a one-dimensional perturbed Hill operator
1986
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T. Kappeler, Solution of the Korteveg-de Vries equation with steplike initial data
1986
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1986
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H. Stephan, Inverse problems for non-decreasing potentials
1986
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E.Ya. Khruslov and V.P. Kotlyarov, Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations
1994
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F. Gesztesy, R. Ratnaseelan, and G. Teschl, The KdV hierarchy and associated trace formulas
1996
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F. Gesztesy, R. Nowell, and W. Pötz, One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics
1997
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E.Ya. Khruslov and H. Stephan, Splitting of some nonlocalized solutions of the Korteveg-de Vries equation into solitons
1998
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A. Boutet de Monvel and I. Egorova, The Toda lattice with step-like initial data. Soliton asymptotics
2000
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B.M. Levitan, Inverse Sturm-Liouville Problems
1987
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N.E. Firsova, Solution of the Cauchy problem for the Korteweg-de Vries equation with initial data that are the sum of a periodic and a rapidly decreasing function
1989
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T.M. Roberts, Scattering for step-periodic potential in one dimension
1990
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T.M. Roberts, Inverse scattering for step-periodic potential in one dimension
1990
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E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro Geometric Approach to Nonlinear Integrable Equations
1994
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F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Volume I: ( 1 + 1 ) (1+1) -Dimensional Continuous Models
2003
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Ag. Kh. Khanmamedov, Transformation operators for the perturbed Hill difference equation and one of their applications
2003
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A. Boutet de Monvel and I. Egorova, Transformation operator for Jacobi matrices with asymptotically periodic coefficients
2004
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I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with quasi-periodic background
2006
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I. Egorova, J. Michor, and G. Teschl, Inverse scattering transform for the Toda hierarchy with quasi-periodic background
2007
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I. Egorova, J. Michor, and G. Teschl, Scattering theory for Jacobi operators with steplike quasi-periodic background
2007
Closest in time.