doi: 10.3934/dcdss.2022111
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Initial-boundary value problems for the two-component complex modified Korteweg-de Vries equation on the interval

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Yi Zhang

This article is dedicated to the Professor Jibin Li on his 80th birthday

Received  December 2021 Revised  April 2022 Early access May 2022

Fund Project: This work is supported by the National Natural Science Foundation of China (No.11371326 and No.11975145)

We apply the Fokas unified method to study initial-boundary value (IBV) problems for the two-component complex modified Korteweg-de Vries (mKdV) equation with a $ 4\times 4 $ Lax pair on the interval. The solution can be written by the solution of a $ 4\times 4 $ Riemann-Hilbert (RH) problem constructed in the complex $ \lambda $-plane. The relevant jump matrices are explicitly expressed in terms of three matrix-valued spectral functions related to the initial values, and the Dirichlet-Neumann boundary values, respectively. Moreover, we get that these spectral functions satisfy a global relation and also study the asymptotic analysis of the spectral functions. By considering the global relation, we express the unknown boundary values in terms of the known initial and boundary values via a Gelfand-Levitan-Marchenko (GLM) representation.

Citation: Rusuo Ye, Yi Zhang. Initial-boundary value problems for the two-component complex modified Korteweg-de Vries equation on the interval. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022111
References:
[1]

M. J. AblowitzD. Bar Yaacov and A. S. Fokas, On the inverse scattering transform for the Kadomtsev-Petviashvili equation, Stud. Appl. Math., 69 (1983), 135-143.  doi: 10.1002/sapm1983692135.

[2]

M. J. AblowitzD. J. KaupA. C. Newell and H. Segur, Method for solving the sine-Gordon equation, Phys. Rev. Lett., 30 (1973), 1262-1264.  doi: 10.1103/PhysRevLett.30.1262.

[3]

D. BilmanR. Buckingham and D.-S. Wang, Far-field asymptotics for multiple-pole solitons in the large-order limit, J. Differential Equations, 297 (2021), 320-369.  doi: 10.1016/j.jde.2021.06.016.

[4]

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. doi: 10.1017/S1474748004000052.

[5]

A. Boutet de MonvelA. S. Fokas and D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Comm. Math. Phys., 263 (2006), 133-172.  doi: 10.1007/s00220-005-1495-2.

[6]

A. Boutet de Monvel and V. Kotlyarov, Scattering problems for the Zakharov-Shabat equations on the semi-axis, Inverse Problems, 16 (2000), 1813-1837.  doi: 10.1088/0266-5611/16/6/314.

[7]

A. Boutet de Monvel and D. Shepelsky, The mKdV equation on a finite interval, C. R. Math. Acad. Sci. Paris, 337 (2003), 517-522.  doi: 10.1016/j.crma.2003.09.009.

[8]

P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Bull. Amer. Math. Soc. (N.S.), 26 (1992), 119-123.  doi: 10.1090/S0273-0979-1992-00253-7.

[9]

P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math., 137 (1993), 295-368.  doi: 10.2307/2946540.

[10]

M. Dieng and K. D. T. R. McLaughlin, Long-time asymptotics for the NLS equation via dbar methods, preprint, 2008, arXiv: 0805.2807.

[11]

N. Flyer and A. S. Fokas, A hybrid analytical-numerical method for solving evolution partial differential equations. I. The half-line, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 1823-1849.  doi: 10.1098/rspa.2008.0041.

[12]

A. S. Fokas, A generalized Dirichlet to Neumann map for certain nonlinear evolution PDEs, Comm. Pure Appl. Math., 58 (2005), 639-670.  doi: 10.1002/cpa.20076.

[13]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, PDEs. Proc. Roy. Soc. London Ser. A, 453 (1997), 1411-1443.  doi: 10.1098/rspa.1997.0077.

[14]

A. S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008. doi: 10.1137/1.9780898717068.

[15]

A. S. Fokas and A. R. Its, The nonlinear Schr$\rm\ddot{o}$dinger equation on the interval, J. Phys. A, 37 (2004), 6091-6114.  doi: 10.1088/0305-4470/37/23/009.

[16]

A. S. Fokas and A. R. Its, An initial-boundary value problem for the sine-Gordon equation, Teoret. Mat. Fiz., 92 (1992), 387-403.  doi: 10.1007/BF01017074.

[17]

A. S. Fokas and A. R. Its, An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simulation, 37 (1994), 293-321.  doi: 10.1016/0378-4754(94)00021-2.

[18]

A. S. FokasA. R. Its and L.-Y. Sung, The nonlinear Schr$\rm\ddot{o}$dinger equation on the half-line, Nonlinearity, 18 (2005), 1771-1822.  doi: 10.1088/0951-7715/18/4/019.

[19]

A. S. Fokas and J. Lenells, The unified method: I. Nonlinearizable problem on the half-line, J. Phys. A, 45 (2012), 195201, 38 pp. doi: 10.1088/1751-8113/45/19/195201.

[20]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Physical Review Letters, 19 (1967), 1095-1097. 

[21]

J. HeL. WangL. LiK. Porsezian and R. Erd$\rm\acute{e}$lyi, Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation, Physical Review E, 89 (2014), 62917. 

[22]

B.-B. HuT.-C. Xia and W.-X. Ma, Riemann-Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line, Appl. Math. Comput., 332 (2018), 148-159.  doi: 10.1016/j.amc.2018.03.049.

[23]

L. HuangJ. Xu and E. Fan, Long-time asymptotic for the Hirota equation via nonlinear steepest descent method, Nonlinear Anal. Real World Appl., 26 (2015), 229-262.  doi: 10.1016/j.nonrwa.2015.05.011.

[24]

A. G. Johnpillai, A. H. Kara and A. Biswas, Exact group invariant solutions and conservation laws of the complex modified Korteweg-de Vries equation, Zeitschrift f$\rm\ddot{u}$r Naturforschung A, 68 (2013), 510–514.

[25]

A. Korkmaz and I. Dag, Numerical simulations of complex modified KdV equation using polynomial differential quadrature method, Int. J. Math. Stat., 10 (2011), 1-13. 

[26]

J. Lenells, The derivative nonlinear Schr$\rm\ddot{o}$dinger equation on the half-line, Phys. D, 237 (2008), 3008-3019.  doi: 10.1016/j.physd.2008.07.005.

[27]

J. Lenells, Initial-boundary value problems for integrable evolution equations with $3\times 3$ Lax pairs, Phys. D, 241 (2012), 857-875.  doi: 10.1016/j.physd.2012.01.010.

[28]

J. Lenells and A. S. Fokas, The unified method: II. NLS on the half-line $t$-periodic boundary conditions, Journal of Physics A: Mathematical and Theoretical, 45 (2012), 195202, 36 pp. doi: 10.1088/1751-8113/45/19/195202.

[29]

J. Lenells and A. S. Fokas, The unified method: III. Nonlinearizable problem on the interval, J. Phys. A, 45 (2012), 95203, 21 pp. doi: 10.1088/1751-8113/45/19/195203.

[30]

H. Liu and X. Geng, Initial-boundary problems for the vector modified Korteweg-de Vries equation via Fokas unified transform method, J. Math. Anal. Appl., 440 (2016), 578-596.  doi: 10.1016/j.jmaa.2016.03.068.

[31]

W.-X. Ma, Long-time asymptotics of a three-component coupled mKdV system, Mathematics, 7 (2019), 573. 

[32]

T. R. Marchant, Asymptotic solitons on a non-zero mean level, Chaos, Solitons and Fractals, 32 (2007), 1328-1336.  doi: 10.1016/j.chaos.2005.11.096.

[33]

K. T. R. McLaughlin and P. D. Miller, The dbar steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, preprint, 2004.

[34]

B. Pelloni, The asymptotic behavior of the solution of boundary value problems for the sine-Gordon equation on a finite interval, J. Nonlinear Math. Phys., 12 (2005), 518-529.  doi: 10.2991/jnmp.2005.12.4.6.

[35]

B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity, 28 (2015), R1–R38. doi: 10.1088/0951-7715/28/2/R1.

[36]

S.-F. Tian, Initial-boundary value problems for the general coupled nonlinear Schr$\rm\ddot{o}$dinger equation on the interval via the Fokas method, J. Differential Equations, 262 (2017), 506-558.  doi: 10.1016/j.jde.2016.09.033.

[37]

S.-F. Tian, Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method, J. Phys. A, 50 (2017), 395204, 32 pp. doi: 10.1088/1751-8121/aa825b.

[38]

S.-F. Tian, Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval, Commun. Pure Appl. Anal., 17 (2018), 923-957.  doi: 10.3934/cpaa.2018046.

[39]

M. Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Japan, 34 (1973), 1289-1296.  doi: 10.1143/JPSJ.34.1289.

[40]

D.-S. WangB. Guo and X. Wang, Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions, J. Differential Equations, 266 (2019), 5209-5253.  doi: 10.1016/j.jde.2018.10.053.

[41]

D.-S. Wang and X. Wang, Long-time asymptotics and the bright $N$-soliton solutions of the Kundu-Eckhaus equation via the Riemann-Hilbert approach, Nonlinear Anal. Real World Appl., 41 (2018), 334-361.  doi: 10.1016/j.nonrwa.2017.10.014.

[42]

F. Wang and W.-X. Ma, A $\bar{\partial}$-steepest descent method for oscillatory Riemann-Hilbert problems, J. Nonlinear Sci., 32 (2022), Paper No. 10, 46 pp. doi: 10.1007/s00332-021-09765-7.

[43]

J. Xu and E. Fan, Long-time asymptotic behavior for the complex short pulse equation, J. Differential Equations, 269 (2020), 10322-10349.  doi: 10.1016/j.jde.2020.07.009.

[44]

J. Xu and E. Fan, The unified transform method for the Sasa-Satsuma equation on the half-line, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20130068, 25 pp. doi: 10.1098/rspa.2013.0068.

[45]

J. Xu and E. Fan, The three-wave equation on the half-line, Phys. Lett. A, 378 (2014), 26-33.  doi: 10.1016/j.physleta.2013.10.027.

[46]

J. Xu, Q. Zhu and E. Fan, The initial-boundary value problem for the Sasa-Satsuma equation on a finite interval via the Fokas method, J. Math. Phys., 59 (2018), 073508, 19 pp. doi: 10.1063/1.5047140.

[47]

Z. Yan, Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a $4\times 4$ Lax pair on a finite interval, Journal of Mathematical Physics, 60 (2019), 083511.  doi: 10.1063/1.5058722.

[48]

Z. Yan, An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a $4\times 4$ Lax pair on the half-line, Chaos, 27 (2017), 053117, 21 pp. doi: 10.1063/1.4984025.

[49]

Z. Yan, An initial-boundary value problem for the general three-component nonlinear Schr$\rm\ddot{o}$dinger equations on a finite interval, IMA J. Appl. Math., 86 (2021), 427-489.  doi: 10.1093/imamat/hxab007.

[50]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69. 

[51]

V. E. Zakharov and A. B. Shabat, Interaction between solitons in a stable medium, Soviet Physics JETP, 37 (1973), 823-828. 

[52]

Y. ZhangR. Ye and W.-X. Ma, Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations, Math. Methods Appl. Sci., 43 (2020), 613-627.  doi: 10.1002/mma.5914.

[53]

Q. ZhuJ. Xu and E. Fan, Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval, J. Nonlinear Math. Phys., 25 (2018), 136-165.  doi: 10.1080/14029251.2018.1440747.

[54]

Q.-Z. ZhuE.-G. Fan and J. Xu, Initial-boundary value problem for two-component Gerdjikov-Ivanov equation with $3\times 3$ Lax pair on half-line, Commun. Theor. Phys. (Beijing), 68 (2017), 425-438.  doi: 10.1088/0253-6102/68/4/425.

show all references

References:
[1]

M. J. AblowitzD. Bar Yaacov and A. S. Fokas, On the inverse scattering transform for the Kadomtsev-Petviashvili equation, Stud. Appl. Math., 69 (1983), 135-143.  doi: 10.1002/sapm1983692135.

[2]

M. J. AblowitzD. J. KaupA. C. Newell and H. Segur, Method for solving the sine-Gordon equation, Phys. Rev. Lett., 30 (1973), 1262-1264.  doi: 10.1103/PhysRevLett.30.1262.

[3]

D. BilmanR. Buckingham and D.-S. Wang, Far-field asymptotics for multiple-pole solitons in the large-order limit, J. Differential Equations, 297 (2021), 320-369.  doi: 10.1016/j.jde.2021.06.016.

[4]

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. doi: 10.1017/S1474748004000052.

[5]

A. Boutet de MonvelA. S. Fokas and D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Comm. Math. Phys., 263 (2006), 133-172.  doi: 10.1007/s00220-005-1495-2.

[6]

A. Boutet de Monvel and V. Kotlyarov, Scattering problems for the Zakharov-Shabat equations on the semi-axis, Inverse Problems, 16 (2000), 1813-1837.  doi: 10.1088/0266-5611/16/6/314.

[7]

A. Boutet de Monvel and D. Shepelsky, The mKdV equation on a finite interval, C. R. Math. Acad. Sci. Paris, 337 (2003), 517-522.  doi: 10.1016/j.crma.2003.09.009.

[8]

P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Bull. Amer. Math. Soc. (N.S.), 26 (1992), 119-123.  doi: 10.1090/S0273-0979-1992-00253-7.

[9]

P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math., 137 (1993), 295-368.  doi: 10.2307/2946540.

[10]

M. Dieng and K. D. T. R. McLaughlin, Long-time asymptotics for the NLS equation via dbar methods, preprint, 2008, arXiv: 0805.2807.

[11]

N. Flyer and A. S. Fokas, A hybrid analytical-numerical method for solving evolution partial differential equations. I. The half-line, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 1823-1849.  doi: 10.1098/rspa.2008.0041.

[12]

A. S. Fokas, A generalized Dirichlet to Neumann map for certain nonlinear evolution PDEs, Comm. Pure Appl. Math., 58 (2005), 639-670.  doi: 10.1002/cpa.20076.

[13]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, PDEs. Proc. Roy. Soc. London Ser. A, 453 (1997), 1411-1443.  doi: 10.1098/rspa.1997.0077.

[14]

A. S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008. doi: 10.1137/1.9780898717068.

[15]

A. S. Fokas and A. R. Its, The nonlinear Schr$\rm\ddot{o}$dinger equation on the interval, J. Phys. A, 37 (2004), 6091-6114.  doi: 10.1088/0305-4470/37/23/009.

[16]

A. S. Fokas and A. R. Its, An initial-boundary value problem for the sine-Gordon equation, Teoret. Mat. Fiz., 92 (1992), 387-403.  doi: 10.1007/BF01017074.

[17]

A. S. Fokas and A. R. Its, An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simulation, 37 (1994), 293-321.  doi: 10.1016/0378-4754(94)00021-2.

[18]

A. S. FokasA. R. Its and L.-Y. Sung, The nonlinear Schr$\rm\ddot{o}$dinger equation on the half-line, Nonlinearity, 18 (2005), 1771-1822.  doi: 10.1088/0951-7715/18/4/019.

[19]

A. S. Fokas and J. Lenells, The unified method: I. Nonlinearizable problem on the half-line, J. Phys. A, 45 (2012), 195201, 38 pp. doi: 10.1088/1751-8113/45/19/195201.

[20]

C. S. GardnerJ. M. GreeneM. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Physical Review Letters, 19 (1967), 1095-1097. 

[21]

J. HeL. WangL. LiK. Porsezian and R. Erd$\rm\acute{e}$lyi, Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation, Physical Review E, 89 (2014), 62917. 

[22]

B.-B. HuT.-C. Xia and W.-X. Ma, Riemann-Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line, Appl. Math. Comput., 332 (2018), 148-159.  doi: 10.1016/j.amc.2018.03.049.

[23]

L. HuangJ. Xu and E. Fan, Long-time asymptotic for the Hirota equation via nonlinear steepest descent method, Nonlinear Anal. Real World Appl., 26 (2015), 229-262.  doi: 10.1016/j.nonrwa.2015.05.011.

[24]

A. G. Johnpillai, A. H. Kara and A. Biswas, Exact group invariant solutions and conservation laws of the complex modified Korteweg-de Vries equation, Zeitschrift f$\rm\ddot{u}$r Naturforschung A, 68 (2013), 510–514.

[25]

A. Korkmaz and I. Dag, Numerical simulations of complex modified KdV equation using polynomial differential quadrature method, Int. J. Math. Stat., 10 (2011), 1-13. 

[26]

J. Lenells, The derivative nonlinear Schr$\rm\ddot{o}$dinger equation on the half-line, Phys. D, 237 (2008), 3008-3019.  doi: 10.1016/j.physd.2008.07.005.

[27]

J. Lenells, Initial-boundary value problems for integrable evolution equations with $3\times 3$ Lax pairs, Phys. D, 241 (2012), 857-875.  doi: 10.1016/j.physd.2012.01.010.

[28]

J. Lenells and A. S. Fokas, The unified method: II. NLS on the half-line $t$-periodic boundary conditions, Journal of Physics A: Mathematical and Theoretical, 45 (2012), 195202, 36 pp. doi: 10.1088/1751-8113/45/19/195202.

[29]

J. Lenells and A. S. Fokas, The unified method: III. Nonlinearizable problem on the interval, J. Phys. A, 45 (2012), 95203, 21 pp. doi: 10.1088/1751-8113/45/19/195203.

[30]

H. Liu and X. Geng, Initial-boundary problems for the vector modified Korteweg-de Vries equation via Fokas unified transform method, J. Math. Anal. Appl., 440 (2016), 578-596.  doi: 10.1016/j.jmaa.2016.03.068.

[31]

W.-X. Ma, Long-time asymptotics of a three-component coupled mKdV system, Mathematics, 7 (2019), 573. 

[32]

T. R. Marchant, Asymptotic solitons on a non-zero mean level, Chaos, Solitons and Fractals, 32 (2007), 1328-1336.  doi: 10.1016/j.chaos.2005.11.096.

[33]

K. T. R. McLaughlin and P. D. Miller, The dbar steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, preprint, 2004.

[34]

B. Pelloni, The asymptotic behavior of the solution of boundary value problems for the sine-Gordon equation on a finite interval, J. Nonlinear Math. Phys., 12 (2005), 518-529.  doi: 10.2991/jnmp.2005.12.4.6.

[35]

B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity, 28 (2015), R1–R38. doi: 10.1088/0951-7715/28/2/R1.

[36]

S.-F. Tian, Initial-boundary value problems for the general coupled nonlinear Schr$\rm\ddot{o}$dinger equation on the interval via the Fokas method, J. Differential Equations, 262 (2017), 506-558.  doi: 10.1016/j.jde.2016.09.033.

[37]

S.-F. Tian, Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method, J. Phys. A, 50 (2017), 395204, 32 pp. doi: 10.1088/1751-8121/aa825b.

[38]

S.-F. Tian, Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval, Commun. Pure Appl. Anal., 17 (2018), 923-957.  doi: 10.3934/cpaa.2018046.

[39]

M. Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Japan, 34 (1973), 1289-1296.  doi: 10.1143/JPSJ.34.1289.

[40]

D.-S. WangB. Guo and X. Wang, Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions, J. Differential Equations, 266 (2019), 5209-5253.  doi: 10.1016/j.jde.2018.10.053.

[41]

D.-S. Wang and X. Wang, Long-time asymptotics and the bright $N$-soliton solutions of the Kundu-Eckhaus equation via the Riemann-Hilbert approach, Nonlinear Anal. Real World Appl., 41 (2018), 334-361.  doi: 10.1016/j.nonrwa.2017.10.014.

[42]

F. Wang and W.-X. Ma, A $\bar{\partial}$-steepest descent method for oscillatory Riemann-Hilbert problems, J. Nonlinear Sci., 32 (2022), Paper No. 10, 46 pp. doi: 10.1007/s00332-021-09765-7.

[43]

J. Xu and E. Fan, Long-time asymptotic behavior for the complex short pulse equation, J. Differential Equations, 269 (2020), 10322-10349.  doi: 10.1016/j.jde.2020.07.009.

[44]

J. Xu and E. Fan, The unified transform method for the Sasa-Satsuma equation on the half-line, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20130068, 25 pp. doi: 10.1098/rspa.2013.0068.

[45]

J. Xu and E. Fan, The three-wave equation on the half-line, Phys. Lett. A, 378 (2014), 26-33.  doi: 10.1016/j.physleta.2013.10.027.

[46]

J. Xu, Q. Zhu and E. Fan, The initial-boundary value problem for the Sasa-Satsuma equation on a finite interval via the Fokas method, J. Math. Phys., 59 (2018), 073508, 19 pp. doi: 10.1063/1.5047140.

[47]

Z. Yan, Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a $4\times 4$ Lax pair on a finite interval, Journal of Mathematical Physics, 60 (2019), 083511.  doi: 10.1063/1.5058722.

[48]

Z. Yan, An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a $4\times 4$ Lax pair on the half-line, Chaos, 27 (2017), 053117, 21 pp. doi: 10.1063/1.4984025.

[49]

Z. Yan, An initial-boundary value problem for the general three-component nonlinear Schr$\rm\ddot{o}$dinger equations on a finite interval, IMA J. Appl. Math., 86 (2021), 427-489.  doi: 10.1093/imamat/hxab007.

[50]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), 62-69. 

[51]

V. E. Zakharov and A. B. Shabat, Interaction between solitons in a stable medium, Soviet Physics JETP, 37 (1973), 823-828. 

[52]

Y. ZhangR. Ye and W.-X. Ma, Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations, Math. Methods Appl. Sci., 43 (2020), 613-627.  doi: 10.1002/mma.5914.

[53]

Q. ZhuJ. Xu and E. Fan, Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval, J. Nonlinear Math. Phys., 25 (2018), 136-165.  doi: 10.1080/14029251.2018.1440747.

[54]

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Figure 1.  The four contours $ \lbrace\gamma_j\rbrace_1^4 $ in the ($ x, t $)-domain
Figure 2.  The domains $ \lbrace D_j\rbrace_1^4 $ in the complex $ \lambda $-plane
Figure 3.  The relations among $ \lbrace\mu_j(x, t, \lambda)\rbrace_{j = 1}^4 $
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