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

  • * Corresponding author: Shou-Fu Tian

    * Corresponding author: Shou-Fu Tian
This work was supported by the Fundamental Research Fund for the Central Universities under the Grant No. 2017XKQY101
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  • 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.

    Mathematics Subject Classification: Primary: 35Q51, 35Q15; Secondary: 41A60.

    Citation:

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  • Figure 1.  The four contours $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ and $\gamma_{4}$ in the $(x, t)-$domain

    Figure 2.  The domains $D_{1}$, $D_{2}$, $D_{3}$ and $D_{4}$ in the complex $k-$plane

  •   M. J. Ablowitz and A. S. Fokas, Introduction and Applications of Complex Variables, Cambridge University Press, second edition, 2003.
      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. 
      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. 
      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.
      A. Constantin , V. S Gerdjikov  and  R. Ivanov , Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006) , 2197-2207. 
      A. Constantin  and  R. Ivanov , Dressing method for the Degasperis-Procesi equation, Stud. Appl. Math., 138 (2017) , 205-226. 
      A. S. Fokas , A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A, 453 (1997) , 1411-1443. 
      A. S. Fokas , Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230 (2002) , 1-39. 
      A. S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008.
      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. 
      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. 
      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. 
      A. S. Fokas  and  A. R. Its , The nonlinear Schrödinger equation on the interval, J. Phys. A: Math. Theor., 37 (2004) , 6091-6114. 
      A. S. Fokas , A. R. Its  and  L. Y. Sung , The nonlinear Schrödinger equation on the half-line, Nonlinearity, 18 (2005) , 1771-1822. 
      A. S. Fokas and J. Lenells, The unified method: Ⅰ. Nonlinearizable problem on the half-line, J. Phys. A: Math. Theor., 45 (2012), 195201.
      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. 
      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. 
      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. 
      R. Hirota , Molecule solutions of coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997) , 2530-2. 
      M. Iwao  and  R. Hirota , Soliton solutions of a coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997) , 577-88. 
      Y. Kurylev  and  M. Lassas , Inverse problems and index formulae for Dirac operators, Adv. Math., 221 (2009) , 170-216. 
      P. D. Lax , Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968) , 467-490. 
      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.
      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.
      J. Lenells and A. S. Fokas, The unified method: Ⅲ. Nonlinearizable problem on the interval, J. Phys. A: Math. Theor., 45 (2012), 195203.
      J. Lenells , Initial-boundary value problems for integrable evolution equations with $3× 3$ Lax pairs, Physica D: Nonlinear Phenomena, 241 (2012) , 857-875. 
      J. Lenells , The Degasperis-Procesi equation on the half-line, Nonlinear Anal., 76 (2013) , 122-139. 
      W. X. Ma  and  R. G. Zhou , Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlinear Math. Phys., 9 (2002) , 106-126. 
      B. Pelloni  and  D. A. Pinotsis , The elliptic sine-Gordon equation in a half plane, Nonlinearity, 23 (2010) , 77-88. 
      B. Pelloni , Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity, 28 (2015) , R1-R38. 
      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. 
      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.
      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.
      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
      T. Tsuchida  and  M. Wadati , The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Japan, 67 (1998) , 1175-1187. 
      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.
      J. Xu  and  E. G. Fan , The three wave equation on the half-line, Phys. Lett. A, 378 (2014) , 26-33. 
      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. 
      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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