Initial boundary value problem for the mKdV equation on a finite interval

Boutet de Monvel, Anne; Shepelsky, Dmitry

doi:10.5802/aif.2056

Initial boundary value problem for the mKdV equation on a finite interval
[Problème aux limites pour l'équation de Korteweg de Vries modifiée sur un intervalle borné]

Boutet de Monvel, Anne ¹ ; Shepelsky, Dmitry

¹ Université Paris 7, Institut de Mathématiques de Jussieu, case 7012, 2 place Jussieu, 75251 Paris (France), Institute for Low Temperature Physics, Mathematical Division, 47 Lenin Avenue, 61103 Kharkov (Ukraine)

Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1477-1495.

Résumé
Abstract

On analyse l’équation de «Korteweg-de Vries modifiée» sur un intervalle borné $(0, L)$ , avec conditions aux limites en $t = 0$ et en $x = 0, L$ , en exprimant sa solution $q (x, t)$ en termes de la solution d’un problème de Riemann-Hilbert associé. Ce problème est défini par des fonctions spectrales déterminées par les conditions aux limites. Nous explicitons la relation globale qui reflète en termes de ces fonctions spectrales la compatibilité des conditions aux limites.

We analyse an initial-boundary value problem for the mKdV equation on a finite interval $(0, L)$ by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex $k$ -plane. This RH problem is determined by certain spectral functions which are defined in terms of the initial-boundary values at $t = 0$ and $x = 0, L$ . We show that the spectral functions satisfy an algebraic “global relation” which express the implicit relation between all boundary values in terms of spectral data.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2056

Classification : 35Q53, 37K15, 35Q15, 34A55, 34L25

Affiliations des auteurs :

Boutet de Monvel, Anne ¹ ; Shepelsky, Dmitry

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Boutet de Monvel, Anne; Shepelsky, Dmitry. Initial boundary value problem for the mKdV equation on a finite interval. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1477-1495. doi : 10.5802/aif.2056. https://www.numdam.org/articles/10.5802/aif.2056/

Bibliographie
Cité par

[1] A. Boutet De Monvel, A.S. Fokas & D. Shepelsky, The mKdV equation on the half-line, J. Inst. Math. Jussieu 3 (2004) p. 139-164 | MR | Zbl

[2] A. Boutet De Monvel, A.S. Fokas & D. Shepelsky, Analysis of the global relation for the nonlinear Schrödinger equation on the half-line, Lett. Math. Phys 65 (2003) p. 199-212 | MR | Zbl

[3] A. Boutet De Monvel & D. Shepelsky, The modified KdV equation on a finite interval, C. R. Math. Acad. Sci. Paris 337 (2003) p. 517-522 | MR | Zbl

[4] A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London, Ser. A 453 (1997) p. 1411-1443 | MR | Zbl

[5] A.S. Fokas, On the integrability of linear and nonlinear partial differential equations, J. Math. Phys 41 (2000) p. 4188-4237 | MR | Zbl

[6] A.S. Fokas, Two dimensional linear PDEs in a convex polygon, Proc. Roy. Soc. London, Ser. A 457 (2001) p. 371-393 | MR | Zbl

[7] A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys 230 (2002) p. 1-39 | MR | Zbl

[8] A.S. Fokas & A.R. Its, The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal 27 (1996) p. 738-764 | MR | Zbl

[9] A.S. Fokas & A.R. Its, The nonlinear Schrödinger equation on the interval, Preprint | MR | Zbl

[10] A.S. Fokas, A.R. Its & L.-Y. Sung, The nonlinear Schrödinger equation on the half-line, Preprint | MR | Zbl

[12] X. Zhou, The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal 20 (1989) p. 966-986 | MR | Zbl

[13] X. Zhou, Inverse scattering transform for systems with rational spectral dependence, J. Differential Equations 115 (1995) p. 277-303 | MR | Zbl

[11] V.E. Zakharov & A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl. 8 (1974) p. 226-235 | Zbl

[<L>11</L>] V.E. Zakharov & A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering. II, Funct. Anal. Appl. 13 (1979) p. 166-174 | Zbl

Zhang, Zechuan; Xu, Taiyang; Fan, Engui Soliton resolution and asymptotic stability of N-soliton solutions for the defocusing mKdV equation with a non-vanishing background, Physica D: Nonlinear Phenomena, Volume 472 (2025), p. 134526 | DOI:10.1016/j.physd.2025.134526
Hu, Jiawei; Zhang, Ning Initial Boundary Value Problem for the Coupled Kundu Equations on the Half-Line, Axioms, Volume 13 (2024) no. 9, p. 579 | DOI:10.3390/axioms13090579
Hu, Jiawei; Zhang, Ning The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line, Symmetry, Volume 16 (2024) no. 10, p. 1258 | DOI:10.3390/sym16101258
Wimmergren, Joseph; Mantzavinos, Dionyssios The linear Lugiato–Lefever equation with forcing and nonzero periodic or nonperiodic boundary conditions, Involve, a Journal of Mathematics, Volume 16 (2023) no. 5, p. 783 | DOI:10.2140/involve.2023.16.783
Ye, Rusuo; Zhang, Yi Two-component complex modified Korteweg–de Vries equations: New soliton solutions from novel binary Darboux transformation, Theoretical and Mathematical Physics, Volume 214 (2023) no. 2, p. 183 | DOI:10.1134/s0040577923020034
Ye, Rusuo; Zhang, Yi Двухкомпонентное комплексное модифицированное уравнение Кортевега-де Фриза: новые солитонные решения, получающиеся из нового бинарного преобразования Дарбу, Теоретическая и математическая физика, Volume 214 (2023) no. 2, p. 211 | DOI:10.4213/tmf10314
Liu, Leilei; Zhang, Weiguo; Xu, Jian; Guo, Yuli The unified transform method to the high-order nonlinear Schrödinger equation with periodic initial condition, Communications in Theoretical Physics, Volume 74 (2022) no. 8, p. 085001 | DOI:10.1088/1572-9494/ac7a23
Li, Ruo-meng; Geng, Xian-guo A Necessary and Sufficient Condition for the Solvability of the Nonlinear Schrödinger Equation on a Finite Interval, Acta Mathematicae Applicatae Sinica, English Series, Volume 37 (2021) no. 1, p. 75 | DOI:10.1007/s10255-021-0994-z
Kalimeris, K. Explicit Soliton Asymptotics for the Nonlinear Schrödinger Equation on the Half-Line, Journal of Nonlinear Mathematical Physics, Volume 17 (2021) no. 4, p. 445 | DOI:10.1142/s1402925110000994
Larkin, Nikolai A.; Luchesi, Jackson Initial-Boundary Value Problems for Nonlinear Dispersive Equations of Higher Orders Posed on Bounded Intervals with General Boundary Conditions, Mathematics, Volume 9 (2021) no. 2, p. 165 | DOI:10.3390/math9020165
Rybalko, Yan; Shepelsky, Dmitry Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation, Physica D: Nonlinear Phenomena, Volume 428 (2021), p. 133060 | DOI:10.1016/j.physd.2021.133060
Yan, Zhenya Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on a finite interval, Journal of Mathematical Physics, Volume 60 (2019) no. 8 | DOI:10.1063/1.5058722
Zhang, Ning; Xia, Tie-Cheng; Hu, Bei-Bei A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line *, Communications in Theoretical Physics, Volume 68 (2017) no. 5, p. 580 | DOI:10.1088/0253-6102/68/5/580
Geng, Xianguo; Liu, Huan; Zhu, Junyi Initial‐Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half‐Line, Studies in Applied Mathematics, Volume 135 (2015) no. 3, p. 310 | DOI:10.1111/sapm.12088
XU, Jian; FAN, Engui A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation, Acta Mathematica Scientia, Volume 34 (2014) no. 4, p. 973 | DOI:10.1016/s0252-9602(14)60063-1
Lenells, Jonatan Initial-boundary value problems for integrable evolution equations with 3×3 Lax pairs, Physica D: Nonlinear Phenomena, Volume 241 (2012) no. 8, p. 857 | DOI:10.1016/j.physd.2012.01.010
Hitzazis, Iasonas; Tsoubelis, Dimitri The Korteweg–de Vries equation on the interval, Journal of Mathematical Physics, Volume 51 (2010) no. 8 | DOI:10.1063/1.3474914
de Monvel, Anne Boutet; Kotlyarov, Vladimir P; Shepelsky, Dmitry; Zheng, Chunxiong Initial boundary value problems for integrable systems: towards the long time asymptotics, Nonlinearity, Volume 23 (2010) no. 10, p. 2483 | DOI:10.1088/0951-7715/23/10/007
Kalimeris, K.; Fokas, A. S. The Heat Equation in the Interior of an Equilateral Triangle, Studies in Applied Mathematics, Volume 124 (2010) no. 3, p. 283 | DOI:10.1111/j.1467-9590.2009.00471.x
Hitzazis, Iasonas; Tsoubelis, Dimitri Riemann–Hilbert formulation for the KdV equation on a finite interval, Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, p. 261 | DOI:10.1016/j.crma.2009.01.012
de Monvel, A. B.; Kotlyarov, V.; Shepelsky, D. Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition, International Mathematics Research Notices (2008) | DOI:10.1093/imrn/rnn139
Boutet de Monvel, Anne; Shepelsky, Dmitry The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach, Journal of Geometric Analysis, Volume 18 (2008) no. 2, p. 285 | DOI:10.1007/s12220-008-9014-2
Doronin, Gleb G.; Larkin, Nikolai A. Boundary value problems for the stationary Kawahara equation, Nonlinear Analysis: Theory, Methods Applications, Volume 69 (2008) no. 5-6, p. 1655 | DOI:10.1016/j.na.2007.07.005
Doronin, Gleb G.; Larkin, Nikolai A. KdV equation in domains with moving boundaries, Journal of Mathematical Analysis and Applications, Volume 328 (2007) no. 1, p. 503 | DOI:10.1016/j.jmaa.2006.05.057
Monvel, Anne Boutet de; Fokas, Athanassis S.; Shepelsky, Dmitry Integrable Nonlinear Evolution Equations on a Finite Interval, Communications in Mathematical Physics, Volume 263 (2006) no. 1, p. 133 | DOI:10.1007/s00220-005-1495-2
Boutet de Monvel, Anne; Shepelsky, Dmitry The Camassa–Holm equation on the half-line, Comptes Rendus. Mathématique, Volume 341 (2005) no. 10, p. 611 | DOI:10.1016/j.crma.2005.09.035

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