American Institute of Mathematical Sciences

May  2018, 17(3): 923-957. doi: 10.3934/cpaa.2018046

Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval

 1 School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, People's Republic of China 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom

* Corresponding author: Shou-Fu Tian

Received  September 2017 Revised  September 2017 Published  January 2018

Fund Project: This work was supported by the Fundamental Research Fund for the Central Universities under the Grant No. 2017XKQY101.

In this paper, we study the initial-boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the finite interval with Lax pairs involving $3× 3$ matrices via the Fokas method. We write the solution in terms of the solution of a $3× 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions $s(k)$, $S(k)$, and $S_{L}(k)$, which are determined by the initial values, boundary values at $x = 0$, and at $x = L$, respectively. Some of the boundary values are known for a well-posed problem, however, the remaining boundary data are unknown. By using the so-called global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data via a Gelfand-Levitan-Marchenko representation.

Citation: Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure and Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
References:
 [1] M. J. Ablowitz and A. S. Fokas, Introduction and Applications of Complex Variables, Cambridge University Press, second edition, 2003. [2] A. Boutet de Monvel, A. S. Fokas and D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Commun. Math. Phys., 263 (2006), 133-172. [3] A. Boutet de Monvel, A. S. Fokas and D. Shepelsky, The mKDV equation on the half-line, J. Inst. Math. Jussieu, 3 (2004), 139-164. [4] G. Biondini and G. Hwang, Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations, Inverse Problems, 24 (2008), 065011. [5] A. Constantin, V. S Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. [6] A. Constantin and R. Ivanov, Dressing method for the Degasperis-Procesi equation, Stud. Appl. Math., 138 (2017), 205-226. [7] A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A, 453 (1997), 1411-1443. [8] A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230 (2002), 1-39. [9] A. S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008. [10] A. S. Fokas and A. R. Its, An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simul., 37 (1994), 293-321. [11] A. R. Fokas and B. Pelloni, The solution of certain initial boundary-value problems for the linearized Korteweg-deVries equation, Proc. R. Soc. Lond. A, 454 (1998), 645-657. [12] A. S. Fokas and A. R. Its, The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal., 27 (1996), 738-764. [13] A. S. Fokas and A. R. Its, The nonlinear Schrödinger equation on the interval, J. Phys. A: Math. Theor., 37 (2004), 6091-6114. [14] A. S. Fokas, A. R. Its and L. Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity, 18 (2005), 1771-1822. [15] A. S. Fokas and J. Lenells, The unified method: Ⅰ. Nonlinearizable problem on the half-line, J. Phys. A: Math. Theor., 45 (2012), 195201. [16] C. S. Gardener, J. M. Greene, M. D. Kruskal and R. M. Miura, Methods for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097. [17] X. G. Geng, H. Liu and J. Zhu, Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line, Stud. Appl. Math., 135 (2015), 310-346. [18] X. G. Geng, Y. Y. Zhai and H. H. Dai, Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math, 263 (2014), 123-153. [19] R. Hirota, Molecule solutions of coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 2530-2. [20] M. Iwao and R. Hirota, Soliton solutions of a coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 577-88. [21] Y. Kurylev and M. Lassas, Inverse problems and index formulae for Dirac operators, Adv. Math., 221 (2009), 170-216. [22] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968), 467-490. [23] J. Lenells and A. S. Fokas, An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons, Inverse problems, 25 (2009), 115006. [24] J. Lenells and A. S. Fokas, The unified method: Ⅱ. NLS on the half-line $t$ -periodic boundary conditions, J. Phys. A: Math. Theor., 45 (2012), 195202. [25] J. Lenells and A. S. Fokas, The unified method: Ⅲ. Nonlinearizable problem on the interval, J. Phys. A: Math. Theor., 45 (2012), 195203. [26] J. Lenells, Initial-boundary value problems for integrable evolution equations with $3× 3$ Lax pairs, Physica D: Nonlinear Phenomena, 241 (2012), 857-875. [27] J. Lenells, The Degasperis-Procesi equation on the half-line, Nonlinear Anal., 76 (2013), 122-139. [28] W. X. Ma and R. G. Zhou, Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlinear Math. Phys., 9 (2002), 106-126. [29] B. Pelloni and D. A. Pinotsis, The elliptic sine-Gordon equation in a half plane, Nonlinearity, 23 (2010), 77-88. [30] B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity, 28 (2015), R1-R38. [31] S. F. Tian, Initial-boundary value problems for the general coupled nonlinear Schrödinger equations on the interval via the Fokas method, J. Differential Equations, 262 (2017), 506-558. [32] S. F. Tian, The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method, Proc. R. Soc. Lond. A, 472 (2016), 20160588. [33] S. F. Tian, Initial-boundary value problemsof the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method, J. Phys. A: Math. Theor., 50 (2017), 395204. [34] S. F. Tian and T. T. Zhang, Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary conditon, Proc. Amer. Math. Soc.. DOI: https://doi.org/10.1090/proc/13917 [35] T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Japan, 67 (1998), 1175-1187. [36] J. Xu and E. G. Fan, The unified transform method for the Sasa-Satsuma equation on the half-line, Proc. R. Soc. London A, 469 (2013), 20130068. [37] J. Xu and E. G. Fan, The three wave equation on the half-line, Phys. Lett. A, 378 (2014), 26-33. [38] J. Xu and E. G. Fan, Initial-boundary value problem for integrable nonlinear evolution equation with $3×3$ Lax pairs on the interval, Stud. Appl. Math., 136 (2016), 321-354. [39] B. Xue, F. Li and G. Yang, Explicit solutions and conservation laws of the coupled modified Korteweg-de Vries equation, Phys. Scr., 90 (2015), 085204.

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References:
 [1] M. J. Ablowitz and A. S. Fokas, Introduction and Applications of Complex Variables, Cambridge University Press, second edition, 2003. [2] A. Boutet de Monvel, A. S. Fokas and D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Commun. Math. Phys., 263 (2006), 133-172. [3] A. Boutet de Monvel, A. S. Fokas and D. Shepelsky, The mKDV equation on the half-line, J. Inst. Math. Jussieu, 3 (2004), 139-164. [4] G. Biondini and G. Hwang, Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations, Inverse Problems, 24 (2008), 065011. [5] A. Constantin, V. S Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. [6] A. Constantin and R. Ivanov, Dressing method for the Degasperis-Procesi equation, Stud. Appl. Math., 138 (2017), 205-226. [7] A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A, 453 (1997), 1411-1443. [8] A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230 (2002), 1-39. [9] A. S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008. [10] A. S. Fokas and A. R. Its, An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simul., 37 (1994), 293-321. [11] A. R. Fokas and B. Pelloni, The solution of certain initial boundary-value problems for the linearized Korteweg-deVries equation, Proc. R. Soc. Lond. A, 454 (1998), 645-657. [12] A. S. Fokas and A. R. Its, The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal., 27 (1996), 738-764. [13] A. S. Fokas and A. R. Its, The nonlinear Schrödinger equation on the interval, J. Phys. A: Math. Theor., 37 (2004), 6091-6114. [14] A. S. Fokas, A. R. Its and L. Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity, 18 (2005), 1771-1822. [15] A. S. Fokas and J. Lenells, The unified method: Ⅰ. Nonlinearizable problem on the half-line, J. Phys. A: Math. Theor., 45 (2012), 195201. [16] C. S. Gardener, J. M. Greene, M. D. Kruskal and R. M. Miura, Methods for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097. [17] X. G. Geng, H. Liu and J. Zhu, Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line, Stud. Appl. Math., 135 (2015), 310-346. [18] X. G. Geng, Y. Y. Zhai and H. H. Dai, Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math, 263 (2014), 123-153. [19] R. Hirota, Molecule solutions of coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 2530-2. [20] M. Iwao and R. Hirota, Soliton solutions of a coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 577-88. [21] Y. Kurylev and M. Lassas, Inverse problems and index formulae for Dirac operators, Adv. Math., 221 (2009), 170-216. [22] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968), 467-490. [23] J. Lenells and A. S. Fokas, An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons, Inverse problems, 25 (2009), 115006. [24] J. Lenells and A. S. Fokas, The unified method: Ⅱ. NLS on the half-line $t$ -periodic boundary conditions, J. Phys. A: Math. Theor., 45 (2012), 195202. [25] J. Lenells and A. S. Fokas, The unified method: Ⅲ. Nonlinearizable problem on the interval, J. Phys. A: Math. Theor., 45 (2012), 195203. [26] J. Lenells, Initial-boundary value problems for integrable evolution equations with $3× 3$ Lax pairs, Physica D: Nonlinear Phenomena, 241 (2012), 857-875. [27] J. Lenells, The Degasperis-Procesi equation on the half-line, Nonlinear Anal., 76 (2013), 122-139. [28] W. X. Ma and R. G. Zhou, Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlinear Math. Phys., 9 (2002), 106-126. [29] B. Pelloni and D. A. Pinotsis, The elliptic sine-Gordon equation in a half plane, Nonlinearity, 23 (2010), 77-88. [30] B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity, 28 (2015), R1-R38. [31] S. F. Tian, Initial-boundary value problems for the general coupled nonlinear Schrödinger equations on the interval via the Fokas method, J. Differential Equations, 262 (2017), 506-558. [32] S. F. Tian, The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method, Proc. R. Soc. Lond. A, 472 (2016), 20160588. [33] S. F. Tian, Initial-boundary value problemsof the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method, J. Phys. A: Math. Theor., 50 (2017), 395204. [34] S. F. Tian and T. T. Zhang, Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary conditon, Proc. Amer. Math. Soc.. DOI: https://doi.org/10.1090/proc/13917 [35] T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Japan, 67 (1998), 1175-1187. [36] J. Xu and E. G. Fan, The unified transform method for the Sasa-Satsuma equation on the half-line, Proc. R. Soc. London A, 469 (2013), 20130068. [37] J. Xu and E. G. Fan, The three wave equation on the half-line, Phys. Lett. A, 378 (2014), 26-33. [38] J. Xu and E. G. Fan, Initial-boundary value problem for integrable nonlinear evolution equation with $3×3$ Lax pairs on the interval, Stud. Appl. Math., 136 (2016), 321-354. [39] B. Xue, F. Li and G. Yang, Explicit solutions and conservation laws of the coupled modified Korteweg-de Vries equation, Phys. Scr., 90 (2015), 085204.
The four contours $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ and $\gamma_{4}$ in the $(x, t)-$domain
The domains $D_{1}$, $D_{2}$, $D_{3}$ and $D_{4}$ in the complex $k-$plane
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