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 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 $\mathbb R$,, , (). 

[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 $\mathbb R$,, , (). 

[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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