One smoothing property of the scattering map of the KdV on \mathbb{R}

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March 2016, 36(3): 1493-1537. doi: 10.3934/dcds.2016.36.1493

One smoothing property of the scattering map of the KdV on $\mathbb{R}$

Alberto Maspero ^1, and Beat Schaad ^2,

1.	Department of Mathematics, Università la Sapienza Roma, Piazzale Aldo Moro, 5, 00185 Rome, Italy
2.	Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Blvd, Lawrence, Kansas 66045-7594, United States

Received December 2014 Revised June 2015 Published August 2015

Abstract
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In this paper we prove that in appropriate weighted Sobolev spaces, in the case of no bound states, the scattering map of the Korteweg-de Vries (KdV) on $\mathbb{R}$ is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV flow and the corresponding Airy flow is 1-smoothing.

Keywords: KdV, scattering theory, spectral theory, integrable systems., smoothing effect.

Mathematics Subject Classification: Primary: 35Q53; Secondary: 35P25, 37K1.

Citation: Alberto Maspero, Beat Schaad. One smoothing property of the scattering map of the KdV on $\mathbb{R}$. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1493-1537. doi: 10.3934/dcds.2016.36.1493

References:

[1]	M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998.
[2]	M. Ablowitz, D. Kaup, A. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math., 53 (1974), 249-315.
[3]	A. Babin, A. Ilyin and E. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356.
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[5]	J. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: 10.1098/rsta.1975.0035.
[6]	A. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1092-1099. doi: 10.1073/pnas.53.5.1092.
[7]	A. Cohen and T. Kappeler, The asymptotic behavior of solutions of the Korteweg-de Vries equation evolving from very irregular data, Ann. Physics, 178 (1987), 144-185. doi: 10.1016/S0003-4916(87)80016-7.
[8]	A. Cohen and T. Kappeler, Solutions to the Korteweg-de Vries equation with initial profile in $L^1_1(R) \cap L^1_N(R^+)$, SIAM J. Math. Anal., 18 (1987), 991-1025. doi: 10.1137/0518076.
[9]	J. Colliander, G. Staffilani and H. Takaoka, Global wellposedness for KDV below $L^2$, Mathematical Research Letters, 6 (1999), 755-778. doi: 10.4310/MRL.1999.v6.n6.a13.
[10]	P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202.
[11]	M. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution, Int. Math. Res. Not. IMRN, 20 (2013), 4589-4614.
[12]	M. Erdoğan and N. Tzirakis, Long time dynamics for forced and weakly damped KdV on the torus, Commun. Pure Appl. Anal., 12 (2013), 2669-2684. doi: 10.3934/cpaa.2013.12.2669.
[13]	L. Faddeev, Properties of the $S$-matrix of the one-dimensional Schrödinger equation, Trudy Mat. Inst. Steklov., 73 (1964), 314-336.
[14]	L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.
[15]	C. Frayer, R. Hryniv, Ya. Mykytyuk and P. Perry, Inverse scattering for Schrödinger operators with Miura potentials. I. Unique Riccati representatives and ZS-AKNS systems, Inverse Problems, 25 (2009), 115007, 25pp. doi: 10.1088/0266-5611/25/11/115007.
[16]	H. Flaschka, On the Toda lattice. II. Inverse-scattering solution, Progr. Theoret. Phys., 51 (1974), 703-716. doi: 10.1143/PTP.51.703.
[17]	C. Gardner, J. Greene, M. Kruskal and R. Miura, Method for Solving the Korteweg-deVries Equation, Physical Review Letters, 19 (1967), 1095-1097.
[18]	C. Gardner, J. Greene, M. Kruskal and R. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math., 27 (1974), 97-133. doi: 10.1002/cpa.3160270108.
[19]	B. Grébert and T. Kappeler, The Defocusing NLS Equation and its Normal Form, European Mathematical Society, Zürich, 2014. doi: 10.4171/131.
[20]	R. Hryniv, Y. Mykytyuk and P. Perry, Sobolev mapping properties of the scattering transform for the Schrödinger equation, in Spectral theory and geometric analysis, 535 (2011), 79-93. doi: 10.1090/conm/535/10536.
[21]	T. Kappeler, Inverse scattering for scattering data with poor regularity or slow decay, J. Integral Equations Appl., 1 (1988), 123-145. doi: 10.1216/JIE-1988-1-1-123.
[22]	T. Kappeler, P. Lohrmann, P. Topalov and N. T. Zung, Birkhoff coordinates for the focusing NLS equation, Comm. Math. Phys., 285 (2009), 1087-1107. doi: 10.1007/s00220-008-0543-0.
[23]	T. Kappeler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.
[24]	T. Kappeler, B. Schaad and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators, in Spectral Geometry, 84, Amer. Math. Soc., Providence, 2012, 243-284. doi: 10.1090/pspum/084/1360.
[25]	T. Kappeler, B. Schaad and P. Topalov, Qualitative features of periodic solutions of KdV, Comm. Partial Differential Equations, 38 (2013), 1626-1673. doi: 10.1080/03605302.2013.814141.
[26]	T. Kappeler and E. Trubowitz, Properties of the scattering map, Comment. Math. Helv., 61 (1986), 442-480. doi: 10.1007/BF02621927.
[27]	T. Kappeler and E. Trubowitz, Properties of the scattering map, II, Comment. Math. Helv., 63 (1988), 150-167. doi: 10.1007/BF02566758.
[28]	T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007/BF01647967.
[29]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in Applied Mathematics, 8, Academic Press, 1983, 93-128.
[30]	C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[31]	S. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions, Geom. Funct. Anal., 20 (2010), 1431-1463. doi: 10.1007/s00039-010-0103-6.
[32]	S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV, Discrete Contin. Dyn. Syst., 27 (2010), 1-24. doi: 10.3934/dcds.2010.27.1.
[33]	V. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986. doi: 10.1007/978-3-0348-5485-6.
[34]	A. Maspero and B. Schaad, One smoothing property of the scattering map of the KdV on $\mathbb R$, arXiv:1412.3268.
[35]	D. McLaughlin, Erratum: Four examples of the inverse method as a canonical transformation, Journal of Mathematical Physics, 16 (1975), 1704-1704.
[36]	D. McLaughlin, Four examples of the inverse method as a canonical transformation, Journal of Mathematical Physics, 16 (1975), 96-99. doi: 10.1063/1.522391.
[37]	J. Mujica, Complex Analysis in Banach Spaces, North-Holland Publishing Co., Amsterdam, 1986.
[38]	J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Comm. Partial Differential Equations, 34 (2009), 1208-1227. doi: 10.1080/03605300903129044.
[39]	S. Novikov, S. Manakov, L. Pitaevskiĭ and V. Zakharov, Theory of Solitons, Consultants Bureau [Plenum], New York, 1984.
[40]	R. Novikov, Inverse scattering up to smooth functions for the Schrödinger equation in dimension $1$, Bull. Sci. Math., 120 (1996), 473-491.
[41]	V. Zakharov and L. Faddeev, Korteweg-de vries equation: A completely integrable hamiltonian system, Functional Analysis and Its Applications, 5 (1971), 280-287. doi: 10.1007/BF01086739.
[42]	V. Zaharov and S. Manakov, The complete integrability of the nonlinear Schrödinger equation, Teoret. Mat. Fiz., 19 (1974), 332-343.
[43]	V. Zakharov and A. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz., 61 (1971), 118-134.
[44]	X. Zhou, $L^2$-Sobolev space bijectivity of the scattering and inverse scattering transforms, Comm. Pure Appl. Math., 51 (1998), 697-731. doi: 10.1002/(SICI)1097-0312(199807)51:7<697::AID-CPA1>3.0.CO;2-1.

show all references

References:

[1]	M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998.
[2]	M. Ablowitz, D. Kaup, A. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math., 53 (1974), 249-315.
[3]	A. Babin, A. Ilyin and E. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., 64 (2011), 591-648. doi: 10.1002/cpa.20356.
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[5]	J. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601. doi: 10.1098/rsta.1975.0035.
[6]	A. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1092-1099. doi: 10.1073/pnas.53.5.1092.
[7]	A. Cohen and T. Kappeler, The asymptotic behavior of solutions of the Korteweg-de Vries equation evolving from very irregular data, Ann. Physics, 178 (1987), 144-185. doi: 10.1016/S0003-4916(87)80016-7.
[8]	A. Cohen and T. Kappeler, Solutions to the Korteweg-de Vries equation with initial profile in $L^1_1(R) \cap L^1_N(R^+)$, SIAM J. Math. Anal., 18 (1987), 991-1025. doi: 10.1137/0518076.
[9]	J. Colliander, G. Staffilani and H. Takaoka, Global wellposedness for KDV below $L^2$, Mathematical Research Letters, 6 (1999), 755-778. doi: 10.4310/MRL.1999.v6.n6.a13.
[10]	P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math., 32 (1979), 121-251. doi: 10.1002/cpa.3160320202.
[11]	M. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution, Int. Math. Res. Not. IMRN, 20 (2013), 4589-4614.
[12]	M. Erdoğan and N. Tzirakis, Long time dynamics for forced and weakly damped KdV on the torus, Commun. Pure Appl. Anal., 12 (2013), 2669-2684. doi: 10.3934/cpaa.2013.12.2669.
[13]	L. Faddeev, Properties of the $S$-matrix of the one-dimensional Schrödinger equation, Trudy Mat. Inst. Steklov., 73 (1964), 314-336.
[14]	L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.
[15]	C. Frayer, R. Hryniv, Ya. Mykytyuk and P. Perry, Inverse scattering for Schrödinger operators with Miura potentials. I. Unique Riccati representatives and ZS-AKNS systems, Inverse Problems, 25 (2009), 115007, 25pp. doi: 10.1088/0266-5611/25/11/115007.
[16]	H. Flaschka, On the Toda lattice. II. Inverse-scattering solution, Progr. Theoret. Phys., 51 (1974), 703-716. doi: 10.1143/PTP.51.703.
[17]	C. Gardner, J. Greene, M. Kruskal and R. Miura, Method for Solving the Korteweg-deVries Equation, Physical Review Letters, 19 (1967), 1095-1097.
[18]	C. Gardner, J. Greene, M. Kruskal and R. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math., 27 (1974), 97-133. doi: 10.1002/cpa.3160270108.
[19]	B. Grébert and T. Kappeler, The Defocusing NLS Equation and its Normal Form, European Mathematical Society, Zürich, 2014. doi: 10.4171/131.
[20]	R. Hryniv, Y. Mykytyuk and P. Perry, Sobolev mapping properties of the scattering transform for the Schrödinger equation, in Spectral theory and geometric analysis, 535 (2011), 79-93. doi: 10.1090/conm/535/10536.
[21]	T. Kappeler, Inverse scattering for scattering data with poor regularity or slow decay, J. Integral Equations Appl., 1 (1988), 123-145. doi: 10.1216/JIE-1988-1-1-123.
[22]	T. Kappeler, P. Lohrmann, P. Topalov and N. T. Zung, Birkhoff coordinates for the focusing NLS equation, Comm. Math. Phys., 285 (2009), 1087-1107. doi: 10.1007/s00220-008-0543-0.
[23]	T. Kappeler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.
[24]	T. Kappeler, B. Schaad and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators, in Spectral Geometry, 84, Amer. Math. Soc., Providence, 2012, 243-284. doi: 10.1090/pspum/084/1360.
[25]	T. Kappeler, B. Schaad and P. Topalov, Qualitative features of periodic solutions of KdV, Comm. Partial Differential Equations, 38 (2013), 1626-1673. doi: 10.1080/03605302.2013.814141.
[26]	T. Kappeler and E. Trubowitz, Properties of the scattering map, Comment. Math. Helv., 61 (1986), 442-480. doi: 10.1007/BF02621927.
[27]	T. Kappeler and E. Trubowitz, Properties of the scattering map, II, Comment. Math. Helv., 63 (1988), 150-167. doi: 10.1007/BF02566758.
[28]	T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007/BF01647967.
[29]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in Applied Mathematics, 8, Academic Press, 1983, 93-128.
[30]	C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[31]	S. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions, Geom. Funct. Anal., 20 (2010), 1431-1463. doi: 10.1007/s00039-010-0103-6.
[32]	S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV, Discrete Contin. Dyn. Syst., 27 (2010), 1-24. doi: 10.3934/dcds.2010.27.1.
[33]	V. Marchenko, Sturm-Liouville Operators and Applications, Birkhäuser, Basel, 1986. doi: 10.1007/978-3-0348-5485-6.
[34]	A. Maspero and B. Schaad, One smoothing property of the scattering map of the KdV on $\mathbb R$, arXiv:1412.3268.
[35]	D. McLaughlin, Erratum: Four examples of the inverse method as a canonical transformation, Journal of Mathematical Physics, 16 (1975), 1704-1704.
[36]	D. McLaughlin, Four examples of the inverse method as a canonical transformation, Journal of Mathematical Physics, 16 (1975), 96-99. doi: 10.1063/1.522391.
[37]	J. Mujica, Complex Analysis in Banach Spaces, North-Holland Publishing Co., Amsterdam, 1986.
[38]	J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Comm. Partial Differential Equations, 34 (2009), 1208-1227. doi: 10.1080/03605300903129044.
[39]	S. Novikov, S. Manakov, L. Pitaevskiĭ and V. Zakharov, Theory of Solitons, Consultants Bureau [Plenum], New York, 1984.
[40]	R. Novikov, Inverse scattering up to smooth functions for the Schrödinger equation in dimension $1$, Bull. Sci. Math., 120 (1996), 473-491.
[41]	V. Zakharov and L. Faddeev, Korteweg-de vries equation: A completely integrable hamiltonian system, Functional Analysis and Its Applications, 5 (1971), 280-287. doi: 10.1007/BF01086739.
[42]	V. Zaharov and S. Manakov, The complete integrability of the nonlinear Schrödinger equation, Teoret. Mat. Fiz., 19 (1974), 332-343.
[43]	V. Zakharov and A. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz., 61 (1971), 118-134.
[44]	X. Zhou, $L^2$-Sobolev space bijectivity of the scattering and inverse scattering transforms, Comm. Pure Appl. Math., 51 (1998), 697-731. doi: 10.1002/(SICI)1097-0312(199807)51:7<697::AID-CPA1>3.0.CO;2-1.

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