Soliton resolution for the focusing modified KdV equation

Chen, Gong; Liu, Jiaqi

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 2005-2071.

The soliton resolution for the focusing modified Korteweg-de Vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation through $\bar{\partial }$-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory $x=\text{v}t$ for any fixed v. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As by-products of our long-time asymptotics, we also obtain the asymptotic stability of nonlinear structures involving solitons and breathers.

[1] Ablowitz, M.; Fokas, A. Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2003 | MR | Zbl

[2] Ablowitz, M.J.; Kaup, D.; Newell, A.C.; Segur, H. The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., Volume 53 (1974) no. 4, pp. 249-315 | DOI | MR | Zbl

[3] Alejo, M.A.; Muñoz, C. Nonlinear stability of mKdV breathers, Commun. Math. Phys., Volume 324 (2013) no. 1, pp. 233-262 | DOI | MR | Zbl

[4] Alejo, M.A.; Muñoz, C. Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers, Anal. PDE, Volume 8 (2015) no. 3, pp. 629-674 | DOI | MR

[5] Beals, R.; Coifman, R.R. Scattering and inverse scattering for first order systems, Commun. Pure Appl. Math., Volume 37 (1984) no. 1, pp. 39-90 | DOI | MR | Zbl

[6] Bona, J.L.; Smith, R. The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A, Volume 278 (1975) no. 1287, pp. 555-601 | DOI | MR | Zbl

[7] Bona, J.L.; Souganidis, P.E.; Strauss, W.A. Stability and instability of solitary waves of Korteweg-de Vries type, Proc. R. Soc. Lond. Ser. A, Volume 411 (1987) no. 1841, pp. 395-412 | DOI | MR | Zbl

[8] Beals, R.; Deift, P.; Tomei, C. Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, vol. 28, American Mathematical Society, Providence, RI, 1988 | MR | Zbl

[9] Borghese, M.; Jenkins, R.; McLaughlin, K.T.-R. Long-time asymptotic behavior of the focusing nonlinear Schrödinger equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 4, pp. 887-920 | DOI | Numdam | MR

[10] Chen, G.; Liu, J. Long-time asymptotics to the modified KdV equation in weighted Sobolev spaces, 2019 (preprint) | arXiv | MR

[11] Christ, M.; Colliander, J.; Tao, T. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Am. J. Math., Volume 125 (2003), pp. 1235-1293 | DOI | MR | Zbl

[12] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$ , J. Am. Math. Soc., Volume 16 (2003) no. 3, pp. 705-749 | DOI | MR | Zbl

[13] Cuccagna, S.; Pelinovsky, D.E. The asymptotic stability of solitons in the cubic NLS equation on the line, Appl. Anal., Volume 93 (2014) no. 4, pp. 791-822 | DOI | MR | Zbl

[14] Deift, P.A. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI, 1999 (viii+273 pp.) | MR | Zbl

[15] Deift, P.; Zhou, X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. Math. (2), Volume 137 (1993), pp. 295-368 | DOI | MR | Zbl

[16] Deift, P.A.; Zhou, X. Asymptotics for the Painlevé II equation, Commun. Pure Appl. Math., Volume 48 (1995) no. 3, pp. 277-337 | DOI | MR | Zbl

[17] Deift, P.; Zhou, X. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser, Commun. Pure Appl. Math., Volume 56 (2003), pp. 1029-1077 | DOI | MR | Zbl

[18] Dieng, M.; McLaughlin, K.D.-T. Long-time asymptotics for the NLS equation via dbar methods, 2008 (preprint) | arXiv

[19] Dieng, M.; McLaughlin, K.D.-T.; Miller, P. Dispersive asymptotics for linear and integrable equations by the $\bar{\partial }$-steepest descent method (preprint) | arXiv | MR

[20] Duyckaerts, T.; Kenig, C.E.; Merle, F. Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math., Volume 1 (2013) no. 1, pp. 75-144 | DOI | MR | Zbl

[21] Duyckaerts, T.; Jia, H.; Kenig, C.E.; Merle, F. Soliton resolution along a sequence of times for the focusing energy critical wave equation, Geom. Funct. Anal., Volume 27 (2017) no. 4, pp. 798-862 | DOI | MR

[22] Germain, P.; Pusateri, F.; Rousset, F. Asymptotic stability of solitons for mKdV, Adv. Math., Volume 299 (2016), pp. 272-330 | DOI | MR

[23] Guo, Z. Global well-posedness of Korteweg-de Vries equation in $H^{-\frac{3}{4}}\left(\mathbb{R}\right)$ , J. Math. Pures Appl. (9), Volume 91 (2009) no. 6, pp. 583-597 | DOI | MR | Zbl

[24] Harrop-Griffiths, B. Long time behavior of solutions to the mKdV, Commun. Partial Differ. Equ., Volume 41 (2016) no. 2, pp. 282-317 | DOI | MR

[25] Hayashi, N.; Naumkin, P. Large time behavior of solutions for the modified Korteweg de Vries equation, Int. Math. Res. Not. (1999), pp. 395-418 | DOI | MR | Zbl

[26] Hayashi, N.; Naumkin, P. On the modified Korteweg-de Vries equation, Math. Phys. Anal. Geom., Volume 4 (2001) no. 3, pp. 197-227 | DOI | MR | Zbl

[27] Hirota, R. Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn., Volume 33 (1972) no. 5, pp. 1456-1458 | DOI

[28] Jenkins, R.; Liu, J.; Perry, P.; Sulem, C. Soliton resolution for the derivative nonlinear Schrödinger equation, Commun. Math. Phys., Volume 363 (2018) no. 3, pp. 1003-1049 | DOI | MR

[29] Kenig, C.; Ponce, G.; Vega, L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Commun. Pure Appl. Math., Volume 46 (1993) no. 4, pp. 527-620 | DOI | MR | Zbl

[30] Kenig, C.; Ponce, G.; Vega, L. On the ill-posedness of some canonical dispersive equations, Duke Math. J., Volume 106 (2001), pp. 617-633 | DOI | MR | Zbl

[31] Killip, R.; Vişan, M.; Zhang, X. Low regularity conservation laws for integrable PDE, Geom. Funct. Anal., Volume 28 (2018) no. 4, pp. 1062-1090 | DOI | MR

[32] Kishimoto, N. Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differ. Integral Equ., Volume 22 (2009) no. 5–6, pp. 447-464 | MR | Zbl

[33] Koch, H.; Tataru, D. Conserved energies for the cubic nonlinear Schrödinger equation in one dimension, Duke Math. J., Volume 167 (2018) no. 17, pp. 3207-3313 | DOI | MR

[34] Lamb, G.L. Elements of Soliton Theory, Pure Appl. Math., Wiley, New York, 1980 | MR | Zbl

[35] Linares, F.; Ponce, G. Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2015 (xiv+301 pp.) | MR | Zbl

[36] Liu, J.; Perry, P.; Sulem, C. Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 1, pp. 217-265 | DOI | Numdam | MR

[37] Martel, Y.; Merle, F. Asymptotic stability of solitons of the subcritical gKdV equations revisited, Nonlinearity, Volume 18 (2005) no. 1, pp. 55-80 | DOI | MR | Zbl

[38] Martel, Y.; Merle, F. Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., Volume 157 (2001) no. 3, pp. 219-254 | DOI | MR | Zbl

[39] Martel, Y.; Merle, F.; Tsai, T.-P. Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Commun. Math. Phys., Volume 231 (2002) no. 2, pp. 347-373 | DOI | MR | Zbl

[40] McLaughlin, K.T.-R.; Miller, P.D. The $\bar{\partial }$ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, Int. Math. Res. Pap. (2006) | MR | Zbl

[41] Mizumachi, T. Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., Volume 32 (2001) no. 5, pp. 1050-1080 | DOI | MR | Zbl

[42] Pego, R.; Weinstein, M.I. Asymptotic stability of solitary waves, Commun. Math. Phys., Volume 164 (1994) no. 2, pp. 305-349 | DOI | MR | Zbl

[43] Pöschel, J.; Trubowitz, E. Inverse Spectral Theory, Pure and Applied Mathematics, vol. 130, Academic Press, Inc., Boston, MA, 1987 (x+192 pp.) | MR | Zbl

[44] Saalmann, A. Long-time asymptotics for the massive Thirring model (preprint) | arXiv

[45] Saut, J.-C. Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. Pures Appl. (9), Volume 58 (1979) no. 1, pp. 21-61 | MR | Zbl

[46] Schuur, P.C. Asymptotic Analysis of Soliton Problems. An Inverse Scattering Approach, Lecture Notes in Mathematics, vol. 1232, Springer-Verlag, Berlin, 1986 (viii+180 pp.) | MR | Zbl

[47] Wadati, M. The modified Korteweg-de Vries equation, J. Phys. Soc. Jpn., Volume 34 (1973) no. 5, pp. 1289-1296 | DOI | MR

[48] Ohkuma, K.; Wadati, M. Multi-pole solutions of the modified Korteweg-de Vries equation, J. Phys. Soc. Jpn., Volume 51 (1982) no. 6, pp. 2029-2035 | DOI | MR

[49] Zhou, X. The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal., Volume 20 (1989) no. 4, pp. 966-986 | DOI | MR | Zbl

[50] Zhou, X. $L^2$-Sobolev space bijectivity of the scattering and inverse scattering transforms, Commun. Pure Appl. Math., Volume 51 (1998), pp. 697-731 | DOI | MR | Zbl