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

Cornelis van der  Mee

doi:10.3934/proc.2015.1089

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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

Abstract
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Under a Creative Commons license

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.
[2]	M.J. Ablowitz, B. Prinari and A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004.
[3]	F. Demontis, Matrix Zakharov-Shabat System and Inverse Scattering Transform, Lambert Academic Publishing, Saarbrücken, 2012.
[4]	F. Demontis and C. van der Mee, Scattering operators for matrix Zakharov-Shabat systems, Integral Equations and Operator Theory, 62 (2008), 517-540.
[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.
[6]	L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, New York, 1987.
[7]	K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962.
[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.
[9]	M. Klaus and C. van der Mee, Wave operators for the matrix Zakharov-Shabat system, J. Mathematical Phys., 51 (2010), 053503.
[10]	V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel and Boston, 1986.
[11]	A. Melin, Operator methods for inverse scattering on the real line, Commun. Partial Differential Equations, 10 (1985), 677-766.
[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.
[13]	C. van der Mee, Nonlinear Evolution Models of Integrable Type, SIMAI e-Lecture Notes 11, SIMAI, Torino, 2013.
[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.
[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.
[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.

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.
[2]	M.J. Ablowitz, B. Prinari and A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004.
[3]	F. Demontis, Matrix Zakharov-Shabat System and Inverse Scattering Transform, Lambert Academic Publishing, Saarbrücken, 2012.
[4]	F. Demontis and C. van der Mee, Scattering operators for matrix Zakharov-Shabat systems, Integral Equations and Operator Theory, 62 (2008), 517-540.
[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.
[6]	L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, New York, 1987.
[7]	K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962.
[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.
[9]	M. Klaus and C. van der Mee, Wave operators for the matrix Zakharov-Shabat system, J. Mathematical Phys., 51 (2010), 053503.
[10]	V.A. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel and Boston, 1986.
[11]	A. Melin, Operator methods for inverse scattering on the real line, Commun. Partial Differential Equations, 10 (1985), 677-766.
[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.
[13]	C. van der Mee, Nonlinear Evolution Models of Integrable Type, SIMAI e-Lecture Notes 11, SIMAI, Torino, 2013.
[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.
[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.
[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.

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