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

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

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

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.
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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