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 & 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.  Google Scholar

[2]

A. Boutet de MonvelA. S. Fokas and D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Commun. Math. Phys., 263 (2006), 133-172.   Google Scholar

[3]

A. Boutet de MonvelA. S. Fokas and D. Shepelsky, The mKDV equation on the half-line, J. Inst. Math. Jussieu, 3 (2004), 139-164.   Google Scholar

[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.  Google Scholar

[5]

A. ConstantinV. S Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.   Google Scholar

[6]

A. Constantin and R. Ivanov, Dressing method for the Degasperis-Procesi equation, Stud. Appl. Math., 138 (2017), 205-226.   Google Scholar

[7]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A, 453 (1997), 1411-1443.   Google Scholar

[8]

A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230 (2002), 1-39.   Google Scholar

[9]

A. S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

A. S. Fokas and A. R. Its, The nonlinear Schrödinger equation on the interval, J. Phys. A: Math. Theor., 37 (2004), 6091-6114.   Google Scholar

[14]

A. S. FokasA. R. Its and L. Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity, 18 (2005), 1771-1822.   Google Scholar

[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.  Google Scholar

[16]

C. S. GardenerJ. M. GreeneM. D. Kruskal and R. M. Miura, Methods for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.   Google Scholar

[17]

X. G. GengH. 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.   Google Scholar

[18]

X. G. GengY. Y. Zhai and H. H. Dai, Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math, 263 (2014), 123-153.   Google Scholar

[19]

R. Hirota, Molecule solutions of coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 2530-2.   Google Scholar

[20]

M. Iwao and R. Hirota, Soliton solutions of a coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 577-88.   Google Scholar

[21]

Y. Kurylev and M. Lassas, Inverse problems and index formulae for Dirac operators, Adv. Math., 221 (2009), 170-216.   Google Scholar

[22]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968), 467-490.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Lenells and A. S. Fokas, The unified method: Ⅲ. Nonlinearizable problem on the interval, J. Phys. A: Math. Theor., 45 (2012), 195203.  Google Scholar

[26]

J. Lenells, Initial-boundary value problems for integrable evolution equations with $3× 3$ Lax pairs, Physica D: Nonlinear Phenomena, 241 (2012), 857-875.   Google Scholar

[27]

J. Lenells, The Degasperis-Procesi equation on the half-line, Nonlinear Anal., 76 (2013), 122-139.   Google Scholar

[28]

W. X. Ma and R. G. Zhou, Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlinear Math. Phys., 9 (2002), 106-126.   Google Scholar

[29]

B. Pelloni and D. A. Pinotsis, The elliptic sine-Gordon equation in a half plane, Nonlinearity, 23 (2010), 77-88.   Google Scholar

[30]

B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity, 28 (2015), R1-R38.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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 Google Scholar

[35]

T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Japan, 67 (1998), 1175-1187.   Google Scholar

[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.  Google Scholar

[37]

J. Xu and E. G. Fan, The three wave equation on the half-line, Phys. Lett. A, 378 (2014), 26-33.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

show all references

References:
[1]

M. J. Ablowitz and A. S. Fokas, Introduction and Applications of Complex Variables, Cambridge University Press, second edition, 2003.  Google Scholar

[2]

A. Boutet de MonvelA. S. Fokas and D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Commun. Math. Phys., 263 (2006), 133-172.   Google Scholar

[3]

A. Boutet de MonvelA. S. Fokas and D. Shepelsky, The mKDV equation on the half-line, J. Inst. Math. Jussieu, 3 (2004), 139-164.   Google Scholar

[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.  Google Scholar

[5]

A. ConstantinV. S Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.   Google Scholar

[6]

A. Constantin and R. Ivanov, Dressing method for the Degasperis-Procesi equation, Stud. Appl. Math., 138 (2017), 205-226.   Google Scholar

[7]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A, 453 (1997), 1411-1443.   Google Scholar

[8]

A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230 (2002), 1-39.   Google Scholar

[9]

A. S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

A. S. Fokas and A. R. Its, The nonlinear Schrödinger equation on the interval, J. Phys. A: Math. Theor., 37 (2004), 6091-6114.   Google Scholar

[14]

A. S. FokasA. R. Its and L. Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity, 18 (2005), 1771-1822.   Google Scholar

[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.  Google Scholar

[16]

C. S. GardenerJ. M. GreeneM. D. Kruskal and R. M. Miura, Methods for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097.   Google Scholar

[17]

X. G. GengH. 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.   Google Scholar

[18]

X. G. GengY. Y. Zhai and H. H. Dai, Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math, 263 (2014), 123-153.   Google Scholar

[19]

R. Hirota, Molecule solutions of coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 2530-2.   Google Scholar

[20]

M. Iwao and R. Hirota, Soliton solutions of a coupled modified KdV equations, J. Phys. Soc. Japan, 66 (1997), 577-88.   Google Scholar

[21]

Y. Kurylev and M. Lassas, Inverse problems and index formulae for Dirac operators, Adv. Math., 221 (2009), 170-216.   Google Scholar

[22]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968), 467-490.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Lenells and A. S. Fokas, The unified method: Ⅲ. Nonlinearizable problem on the interval, J. Phys. A: Math. Theor., 45 (2012), 195203.  Google Scholar

[26]

J. Lenells, Initial-boundary value problems for integrable evolution equations with $3× 3$ Lax pairs, Physica D: Nonlinear Phenomena, 241 (2012), 857-875.   Google Scholar

[27]

J. Lenells, The Degasperis-Procesi equation on the half-line, Nonlinear Anal., 76 (2013), 122-139.   Google Scholar

[28]

W. X. Ma and R. G. Zhou, Adjoint symmetry constraints leading to binary nonlinearization, J. Nonlinear Math. Phys., 9 (2002), 106-126.   Google Scholar

[29]

B. Pelloni and D. A. Pinotsis, The elliptic sine-Gordon equation in a half plane, Nonlinearity, 23 (2010), 77-88.   Google Scholar

[30]

B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity, 28 (2015), R1-R38.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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 Google Scholar

[35]

T. Tsuchida and M. Wadati, The coupled modified Korteweg-de Vries equations, J. Phys. Soc. Japan, 67 (1998), 1175-1187.   Google Scholar

[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.  Google Scholar

[37]

J. Xu and E. G. Fan, The three wave equation on the half-line, Phys. Lett. A, 378 (2014), 26-33.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

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
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