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

Alberto  Maspero; Beat  Schaad

doi:10.3934/dcds.2016.36.1493

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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 & Continuous Dynamical Systems - A, 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, (1991). doi: 10.1017/CBO9780511623998. Google Scholar
[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. Google Scholar
[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. doi: 10.1002/cpa.20356. Google Scholar
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011). Google Scholar
[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. doi: 10.1098/rsta.1975.0035. Google Scholar
[6]	A. Calderón, Commutators of singular integral operators,, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1092. doi: 10.1073/pnas.53.5.1092. Google Scholar
[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. doi: 10.1016/S0003-4916(87)80016-7. Google Scholar
[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. doi: 10.1137/0518076. Google Scholar
[9]	J. Colliander, G. Staffilani and H. Takaoka, Global wellposedness for KDV below $L^2$,, Mathematical Research Letters, 6 (1999), 755. doi: 10.4310/MRL.1999.v6.n6.a13. Google Scholar
[10]	P. Deift and E. Trubowitz, Inverse scattering on the line,, Comm. Pure Appl. Math., 32 (1979), 121. doi: 10.1002/cpa.3160320202. Google Scholar
[11]	M. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, Int. Math. Res. Not. IMRN, 20 (2013), 4589. Google Scholar
[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. doi: 10.3934/cpaa.2013.12.2669. Google Scholar
[13]	L. Faddeev, Properties of the $S$-matrix of the one-dimensional Schrödinger equation,, Trudy Mat. Inst. Steklov., 73 (1964), 314. Google Scholar
[14]	L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons,, Springer Series in Soviet Mathematics, (1987). doi: 10.1007/978-3-540-69969-9. Google Scholar
[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). doi: 10.1088/0266-5611/25/11/115007. Google Scholar
[16]	H. Flaschka, On the Toda lattice. II. Inverse-scattering solution,, Progr. Theoret. Phys., 51 (1974), 703. doi: 10.1143/PTP.51.703. Google Scholar
[17]	C. Gardner, J. Greene, M. Kruskal and R. Miura, Method for Solving the Korteweg-deVries Equation,, Physical Review Letters, 19 (1967), 1095. Google Scholar
[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. doi: 10.1002/cpa.3160270108. Google Scholar
[19]	B. Grébert and T. Kappeler, The Defocusing NLS Equation and its Normal Form,, European Mathematical Society, (2014). doi: 10.4171/131. Google Scholar
[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. doi: 10.1090/conm/535/10536. Google Scholar
[21]	T. Kappeler, Inverse scattering for scattering data with poor regularity or slow decay,, J. Integral Equations Appl., 1 (1988), 123. doi: 10.1216/JIE-1988-1-1-123. Google Scholar
[22]	T. Kappeler, P. Lohrmann, P. Topalov and N. T. Zung, Birkhoff coordinates for the focusing NLS equation,, Comm. Math. Phys., 285 (2009), 1087. doi: 10.1007/s00220-008-0543-0. Google Scholar
[23]	T. Kappeler and J. Pöschel, KdV & KAM,, Springer-Verlag, (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar
[24]	T. Kappeler, B. Schaad and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators,, in Spectral Geometry, (2012), 243. doi: 10.1090/pspum/084/1360. Google Scholar
[25]	T. Kappeler, B. Schaad and P. Topalov, Qualitative features of periodic solutions of KdV,, Comm. Partial Differential Equations, 38 (2013), 1626. doi: 10.1080/03605302.2013.814141. Google Scholar
[26]	T. Kappeler and E. Trubowitz, Properties of the scattering map,, Comment. Math. Helv., 61 (1986), 442. doi: 10.1007/BF02621927. Google Scholar
[27]	T. Kappeler and E. Trubowitz, Properties of the scattering map, II,, Comment. Math. Helv., 63 (1988), 150. doi: 10.1007/BF02566758. Google Scholar
[28]	T. Kato, On the Korteweg-de Vries equation,, Manuscripta Math., 28 (1979), 89. doi: 10.1007/BF01647967. Google Scholar
[29]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, in Studies in Applied Mathematics, (1983), 93. Google Scholar
[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. doi: 10.1002/cpa.3160460405. Google Scholar
[31]	S. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions,, Geom. Funct. Anal., 20 (2010), 1431. doi: 10.1007/s00039-010-0103-6. Google Scholar
[32]	S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV,, Discrete Contin. Dyn. Syst., 27 (2010), 1. doi: 10.3934/dcds.2010.27.1. Google Scholar
[33]	V. Marchenko, Sturm-Liouville Operators and Applications,, Birkhäuser, (1986). doi: 10.1007/978-3-0348-5485-6. Google Scholar
[34]	A. Maspero and B. Schaad, One smoothing property of the scattering map of the KdV on $\mathbb R$,, , (). Google Scholar
[35]	D. McLaughlin, Erratum: Four examples of the inverse method as a canonical transformation,, Journal of Mathematical Physics, 16 (1975), 1704. Google Scholar
[36]	D. McLaughlin, Four examples of the inverse method as a canonical transformation,, Journal of Mathematical Physics, 16 (1975), 96. doi: 10.1063/1.522391. Google Scholar
[37]	J. Mujica, Complex Analysis in Banach Spaces,, North-Holland Publishing Co., (1986). Google Scholar
[38]	J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation,, Comm. Partial Differential Equations, 34 (2009), 1208. doi: 10.1080/03605300903129044. Google Scholar
[39]	S. Novikov, S. Manakov, L. Pitaevskiĭ and V. Zakharov, Theory of Solitons,, Consultants Bureau [Plenum], (1984). Google Scholar
[40]	R. Novikov, Inverse scattering up to smooth functions for the Schrödinger equation in dimension $1$,, Bull. Sci. Math., 120 (1996), 473. Google Scholar
[41]	V. Zakharov and L. Faddeev, Korteweg-de vries equation: A completely integrable hamiltonian system,, Functional Analysis and Its Applications, 5 (1971), 280. doi: 10.1007/BF01086739. Google Scholar
[42]	V. Zaharov and S. Manakov, The complete integrability of the nonlinear Schrödinger equation,, Teoret. Mat. Fiz., 19 (1974), 332. Google Scholar
[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. Google Scholar
[44]	X. Zhou, $L^2$-Sobolev space bijectivity of the scattering and inverse scattering transforms,, Comm. Pure Appl. Math., 51 (1998), 697. doi: 10.1002/(SICI)1097-0312(199807)51:7<697::AID-CPA1>3.0.CO;2-1. Google Scholar

show all references

References:

[1]	M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511623998. Google Scholar
[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. Google Scholar
[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. doi: 10.1002/cpa.20356. Google Scholar
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011). Google Scholar
[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. doi: 10.1098/rsta.1975.0035. Google Scholar
[6]	A. Calderón, Commutators of singular integral operators,, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1092. doi: 10.1073/pnas.53.5.1092. Google Scholar
[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. doi: 10.1016/S0003-4916(87)80016-7. Google Scholar
[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. doi: 10.1137/0518076. Google Scholar
[9]	J. Colliander, G. Staffilani and H. Takaoka, Global wellposedness for KDV below $L^2$,, Mathematical Research Letters, 6 (1999), 755. doi: 10.4310/MRL.1999.v6.n6.a13. Google Scholar
[10]	P. Deift and E. Trubowitz, Inverse scattering on the line,, Comm. Pure Appl. Math., 32 (1979), 121. doi: 10.1002/cpa.3160320202. Google Scholar
[11]	M. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, Int. Math. Res. Not. IMRN, 20 (2013), 4589. Google Scholar
[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. doi: 10.3934/cpaa.2013.12.2669. Google Scholar
[13]	L. Faddeev, Properties of the $S$-matrix of the one-dimensional Schrödinger equation,, Trudy Mat. Inst. Steklov., 73 (1964), 314. Google Scholar
[14]	L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons,, Springer Series in Soviet Mathematics, (1987). doi: 10.1007/978-3-540-69969-9. Google Scholar
[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). doi: 10.1088/0266-5611/25/11/115007. Google Scholar
[16]	H. Flaschka, On the Toda lattice. II. Inverse-scattering solution,, Progr. Theoret. Phys., 51 (1974), 703. doi: 10.1143/PTP.51.703. Google Scholar
[17]	C. Gardner, J. Greene, M. Kruskal and R. Miura, Method for Solving the Korteweg-deVries Equation,, Physical Review Letters, 19 (1967), 1095. Google Scholar
[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. doi: 10.1002/cpa.3160270108. Google Scholar
[19]	B. Grébert and T. Kappeler, The Defocusing NLS Equation and its Normal Form,, European Mathematical Society, (2014). doi: 10.4171/131. Google Scholar
[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. doi: 10.1090/conm/535/10536. Google Scholar
[21]	T. Kappeler, Inverse scattering for scattering data with poor regularity or slow decay,, J. Integral Equations Appl., 1 (1988), 123. doi: 10.1216/JIE-1988-1-1-123. Google Scholar
[22]	T. Kappeler, P. Lohrmann, P. Topalov and N. T. Zung, Birkhoff coordinates for the focusing NLS equation,, Comm. Math. Phys., 285 (2009), 1087. doi: 10.1007/s00220-008-0543-0. Google Scholar
[23]	T. Kappeler and J. Pöschel, KdV & KAM,, Springer-Verlag, (2003). doi: 10.1007/978-3-662-08054-2. Google Scholar
[24]	T. Kappeler, B. Schaad and P. Topalov, Asymptotics of spectral quantities of Schrödinger operators,, in Spectral Geometry, (2012), 243. doi: 10.1090/pspum/084/1360. Google Scholar
[25]	T. Kappeler, B. Schaad and P. Topalov, Qualitative features of periodic solutions of KdV,, Comm. Partial Differential Equations, 38 (2013), 1626. doi: 10.1080/03605302.2013.814141. Google Scholar
[26]	T. Kappeler and E. Trubowitz, Properties of the scattering map,, Comment. Math. Helv., 61 (1986), 442. doi: 10.1007/BF02621927. Google Scholar
[27]	T. Kappeler and E. Trubowitz, Properties of the scattering map, II,, Comment. Math. Helv., 63 (1988), 150. doi: 10.1007/BF02566758. Google Scholar
[28]	T. Kato, On the Korteweg-de Vries equation,, Manuscripta Math., 28 (1979), 89. doi: 10.1007/BF01647967. Google Scholar
[29]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, in Studies in Applied Mathematics, (1983), 93. Google Scholar
[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. doi: 10.1002/cpa.3160460405. Google Scholar
[31]	S. Kuksin, Damped-driven KdV and effective equations for long-time behaviour of its solutions,, Geom. Funct. Anal., 20 (2010), 1431. doi: 10.1007/s00039-010-0103-6. Google Scholar
[32]	S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV,, Discrete Contin. Dyn. Syst., 27 (2010), 1. doi: 10.3934/dcds.2010.27.1. Google Scholar
[33]	V. Marchenko, Sturm-Liouville Operators and Applications,, Birkhäuser, (1986). doi: 10.1007/978-3-0348-5485-6. Google Scholar
[34]	A. Maspero and B. Schaad, One smoothing property of the scattering map of the KdV on $\mathbb R$,, , (). Google Scholar
[35]	D. McLaughlin, Erratum: Four examples of the inverse method as a canonical transformation,, Journal of Mathematical Physics, 16 (1975), 1704. Google Scholar
[36]	D. McLaughlin, Four examples of the inverse method as a canonical transformation,, Journal of Mathematical Physics, 16 (1975), 96. doi: 10.1063/1.522391. Google Scholar
[37]	J. Mujica, Complex Analysis in Banach Spaces,, North-Holland Publishing Co., (1986). Google Scholar
[38]	J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation,, Comm. Partial Differential Equations, 34 (2009), 1208. doi: 10.1080/03605300903129044. Google Scholar
[39]	S. Novikov, S. Manakov, L. Pitaevskiĭ and V. Zakharov, Theory of Solitons,, Consultants Bureau [Plenum], (1984). Google Scholar
[40]	R. Novikov, Inverse scattering up to smooth functions for the Schrödinger equation in dimension $1$,, Bull. Sci. Math., 120 (1996), 473. Google Scholar
[41]	V. Zakharov and L. Faddeev, Korteweg-de vries equation: A completely integrable hamiltonian system,, Functional Analysis and Its Applications, 5 (1971), 280. doi: 10.1007/BF01086739. Google Scholar
[42]	V. Zaharov and S. Manakov, The complete integrability of the nonlinear Schrödinger equation,, Teoret. Mat. Fiz., 19 (1974), 332. Google Scholar
[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. Google Scholar
[44]	X. Zhou, $L^2$-Sobolev space bijectivity of the scattering and inverse scattering transforms,, Comm. Pure Appl. Math., 51 (1998), 697. doi: 10.1002/(SICI)1097-0312(199807)51:7<697::AID-CPA1>3.0.CO;2-1. Google Scholar

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