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2015, 2015(special): 1089-1097. doi: 10.3934/proc.2015.1089

Direct scattering of AKNS systems with $L^2$ potentials

Cornelis van der Mee 1,

1. 

Dip. Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09123 Cagliari, Italy

Received  September 2014 Revised  February 2015 Published  November 2015

In this article the Jost solutions of the AKNS system with suitably weighted $L^2$ potential are constructed as Hardy space perturbations of their space-infinity asymptotics. The reflection coefficients are proven to be $L^2$-functions when the transmission coefficients are $L^\infty$-functions.
Keywords: AKNS system, characterization of scattering data..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Cornelis van der Mee. Direct scattering of AKNS systems with $L^2$ potentials. Conference Publications, 2015, 2015 (special) : 1089-1097. doi: 10.3934/proc.2015.1089
References:
[1]

M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems, Studies in Appl. Math., 53 (1974), 249-315. Google Scholar

[2]

M.J. Ablowitz, B. Prinari and A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004. Google Scholar

[3]

F. Demontis, Matrix Zakharov-Shabat System and Inverse Scattering Transform, Lambert Academic Publishing, Saarbrücken, 2012. Google Scholar

[4]

F. Demontis and C. van der Mee, Scattering operators for matrix Zakharov-Shabat systems, Integral Equations and Operator Theory, 62 (2008), 517-540. Google Scholar

[5]

F. Demontis and C. van der Mee, Characterization of scattering data for the matrix Zakharov-Shabat system, Acta Appl. Math., 131 (2014), 29-47. Google Scholar

[6]

L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, New York, 1987. Google Scholar

[7]

K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. Google Scholar

[8]

M. Klaus, On the eigenvalues of the Lax operator for the matrix-valued AKNS system, in Topics in Operator Theory. II. Systems and Mathematical Physics, Birkhäuser, Basel, 2010. Google Scholar

[9]

M. Klaus and C. van der Mee, Wave operators for the matrix Zakharov-Shabat system, J. Mathematical Phys., 51 (2010), 053503. Google Scholar

[10]

V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel and Boston, 1986. Google Scholar

[11]

A. Melin, Operator methods for inverse scattering on the real line, Commun. Partial Differential Equations, 10 (1985), 677-766. Google Scholar

[12]

S.P. Novikov, S.V. Manakov, L.B. Pitaevskii and V.E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Plenum Press, New York, 1984. Google Scholar

[13]

C. van der Mee, Nonlinear Evolution Models of Integrable Type, SIMAI e-Lecture Notes 11, SIMAI, Torino, 2013. Google Scholar

[14]

C. van der Mee, Time-evolution-proof scattering data for the focusing and defocusing Zakharov-Shabat systems, J. Nonlinear Math. Phys., 21 (2014), 265-277. Google Scholar

[15]

J. Villarroel, M.J. Ablowitz and B. Prinari, Solvability of the direct and inverse problems for the nonlinear Schrödinger equation, Acta Appl. Math., 87 (2005), 245-280. Google Scholar

[16]

V. E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69. Google Scholar

show all references

References:
[1]

M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems, Studies in Appl. Math., 53 (1974), 249-315. Google Scholar

[2]

M.J. Ablowitz, B. Prinari and A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004. Google Scholar

[3]

F. Demontis, Matrix Zakharov-Shabat System and Inverse Scattering Transform, Lambert Academic Publishing, Saarbrücken, 2012. Google Scholar

[4]

F. Demontis and C. van der Mee, Scattering operators for matrix Zakharov-Shabat systems, Integral Equations and Operator Theory, 62 (2008), 517-540. Google Scholar

[5]

F. Demontis and C. van der Mee, Characterization of scattering data for the matrix Zakharov-Shabat system, Acta Appl. Math., 131 (2014), 29-47. Google Scholar

[6]

L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, New York, 1987. Google Scholar

[7]

K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. Google Scholar

[8]

M. Klaus, On the eigenvalues of the Lax operator for the matrix-valued AKNS system, in Topics in Operator Theory. II. Systems and Mathematical Physics, Birkhäuser, Basel, 2010. Google Scholar

[9]

M. Klaus and C. van der Mee, Wave operators for the matrix Zakharov-Shabat system, J. Mathematical Phys., 51 (2010), 053503. Google Scholar

[10]

V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel and Boston, 1986. Google Scholar

[11]

A. Melin, Operator methods for inverse scattering on the real line, Commun. Partial Differential Equations, 10 (1985), 677-766. Google Scholar

[12]

S.P. Novikov, S.V. Manakov, L.B. Pitaevskii and V.E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Plenum Press, New York, 1984. Google Scholar

[13]

C. van der Mee, Nonlinear Evolution Models of Integrable Type, SIMAI e-Lecture Notes 11, SIMAI, Torino, 2013. Google Scholar

[14]

C. van der Mee, Time-evolution-proof scattering data for the focusing and defocusing Zakharov-Shabat systems, J. Nonlinear Math. Phys., 21 (2014), 265-277. Google Scholar

[15]

J. Villarroel, M.J. Ablowitz and B. Prinari, Solvability of the direct and inverse problems for the nonlinear Schrödinger equation, Acta Appl. Math., 87 (2005), 245-280. Google Scholar

[16]

V. E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69. Google Scholar

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