×

The Fokas method for the Broer-Kaup system on the half-line. (English) Zbl 1497.35331

Summary: We analyze the Broer-Kaup system posed on the half-line by using the unified transform method, also known as the Fokas method. We derive the formal representation of the solution for the Broer-Kaup system in terms of the solution of the matrix Riemann-Hilbert problem formulated in the complex plane of the spectral parameter. The jump matrix is uniquely defined by the spectral functions that satisfy a certain relation, called the global relation involving the initial and boundary values. Furthermore, the spectral functions constructed from the initial values and the boundary values are investigated, plus their associated Riemann-Hilbert problems as the inverse problems.

MSC:

35Q15 Riemann-Hilbert problems in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Broer, LJF, Approximate equations for long water waves, Appl. Sci. Res., 31, 377-395 (1975) · Zbl 0326.76017 · doi:10.1007/BF00418048
[2] Kaup, DJ, A higher-order water-wave equation and the method for solving it, Prog. Theor. Phys., 54, 396-408 (1975) · Zbl 1079.37514 · doi:10.1143/PTP.54.396
[3] Hirota, R., Solutions of the classical Boussinesq equation and spherical Boussinesq equation: the Wronskian technique, J. Phys. Soc. Jpn., 55, 2137-2150 (1986) · doi:10.1143/JPSJ.55.2137
[4] Wang, X.; Zhu, J., Broer-Kaup system with corrections via inverse scattering transform, Avd. Math. Phys., 2020, 7859897 (2020) · Zbl 1478.35091
[5] Zhu, J.; Wang, X., Broer-Kaup system revisit: inelastic interaction and blowup solutions, J. Math. Anal. Appl., 496, 124794 (2021) · Zbl 1459.35052 · doi:10.1016/j.jmaa.2020.124794
[6] Ying, JP; Lou, SY, Abundant coherent structures of the (2+1)-dimensional Borer-Kaup-Kupershmidt equation, A. Naturforsch., 53, 251-258 (1998) · doi:10.1515/zna-1998-0523
[7] Kaup, DJ, A perturbation expansion for the Zakharov-Shabat inverse scattering transform, SIAM J. Appl. Math., 31, 121-133 (1976) · Zbl 0334.47006 · doi:10.1137/0131013
[8] Li, YS; Ma, WX; Zhang, JE, Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A, 275, 60-66 (2000) · Zbl 1115.35329 · doi:10.1016/S0375-9601(00)00583-1
[9] Kupershmidt, BA, Mathematics of dispersive water waves, Commun. Math. Phys., 99, 51-73 (1985) · Zbl 1093.37511 · doi:10.1007/BF01466593
[10] Satsuma, J.; Kajiwara, K.; Matsukidaira, J.; Hietarinta, J., Solutions of the system through its trilinear form, J. Phys. Soc. Jpn., 61, 3096-3102 (1992) · Zbl 0941.37547 · doi:10.1143/JPSJ.61.3096
[11] Meng, Q.; Li, W.; He, B., Smooth and peaked solitary wave solutions of the Broer-Kaup system using the approach of dynamical system, Commun. Theor. Phys., 62, 308-314 (2014) · Zbl 1298.35163 · doi:10.1088/0253-6102/62/3/03
[12] Jiang, B.; Bi, QS, Peaked periodic wave solutions to the Broer-Kaup equation, Commun. Theor. Phys., 67, 22-26 (2017) · Zbl 1357.76010 · doi:10.1088/0253-6102/67/1/22
[13] Fokas, AS, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. Lond. A, 453, 1411-1443 (1997) · Zbl 0876.35102 · doi:10.1098/rspa.1997.0077
[14] Fokas, A. S.: A Unified approach to boundary value problems, (CBMS-NSF Regional Conference Series in Applied Mathematics) Philadelphia, SIAM (2008) · Zbl 1181.35002
[15] Fokas, AS, Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230, 1-39 (2002) · Zbl 1010.35089 · doi:10.1007/s00220-002-0681-8
[16] Lenells, J., The derivative nonlinear Schrödinger equation on the half-line, Phys. D, 237, 3008-3019 (2008) · Zbl 1161.35503 · doi:10.1016/j.physd.2008.07.005
[17] Fokas, AS, A new transform method for evolution partial differential equations, IMA J. Appl. Math., 67, 559-590 (2002) · Zbl 1028.35009 · doi:10.1093/imamat/67.6.559
[18] Deconinck, B.; Trogdon, T.; Vasan, V., The method of Fokas for solving linear partial differential equations, SIAM Rev., 56, 159-186 (2014) · Zbl 1295.35002 · doi:10.1137/110821871
[19] Pelloni, B.; Pinotsis, DA, The elliptic sine-Gordon equation in a half plane, Nonlinearity, 23, 77-88 (2010) · Zbl 1182.35108 · doi:10.1088/0951-7715/23/1/004
[20] Colbrook, MJ; Flyer, N.; Fornberg, B., On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains, J. Comput. Phys., 374, 996-1016 (2018) · Zbl 1416.65472 · doi:10.1016/j.jcp.2018.08.005
[21] Hwang, G., The elliptic sinh-Gordon equation in a semi-strip, Adv. Nonlinear Anal., 8, 533-544 (2019) · Zbl 1499.35419 · doi:10.1515/anona-2016-0206
[22] Biondini, G.; Hwang, G., Initial-boundary value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations, Inv. Probl., 24, 65011, 1-44 (2008) · Zbl 1157.35115
[23] Moon, B.; Hwang, G., Discrete linear evolution equations in a finite lattice, J. Differ. Equ. Appl., 25, 630-646 (2019) · Zbl 1421.34014 · doi:10.1080/10236198.2019.1613386
[24] Tian, SF, The mixed coupled nonlinear Schrodinger equation on the half-line via the Fokas method, Proc. R. Soc. A, 472, 20160588 (2016) · Zbl 1371.35278 · doi:10.1098/rspa.2016.0588
[25] Tian, SF, Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method, J. Differ. Equ., 262, 506-558 (2017) · Zbl 1432.35194 · doi:10.1016/j.jde.2016.09.033
[26] Tian, SF, Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval, Commun. Pure Appl. Anal., 17, 923-957 (2018) · Zbl 1397.35262 · doi:10.3934/cpaa.2018046
[27] Fokas, AS, The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs, Comm. Pure Appl. Math. LVII, I, 639-670 (2005) · Zbl 1092.35102 · doi:10.1002/cpa.20076
[28] Fokas, AS; Pelloni, B., The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation, Nonlinearity, 25, 1011-1031 (2012) · Zbl 1241.35069 · doi:10.1088/0951-7715/25/4/1011
[29] Hwang, G., The Fokas method: the Dirichlet to Neumann map for the sine-Gordon equation, Stud. Appl. Math., 132, 381-406 (2014) · Zbl 1301.35133 · doi:10.1111/sapm.12035
[30] Crowdy, DG; Luca, E., Solving Wiener-Hopf problems without kernel factorization, Proc. R. Soc. A, 470, 20140304 (2014) · Zbl 1371.45002 · doi:10.1098/rspa.2014.0304
[31] Colbrook, MJ; Ayton, LJ; Fokas, AS, The unified transform for mixed boundary condition problems in unbounded domains, Proc. R. Soc. A, 475, 20180605 (2019) · Zbl 1472.74115 · doi:10.1098/rspa.2018.0605
[32] Deift, P.; Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems, Bull. Am. Math. Soc., 26, 119-123 (1992) · Zbl 0746.35031 · doi:10.1090/S0273-0979-1992-00253-7
[33] Deift, P.; Venakides, S.; Zhou, X., New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Int. Math. Res. Not., 6, 286-299 (1997) · Zbl 0873.65111 · doi:10.1155/S1073792897000214
[34] Lenells, J.; Fokas, AS, The unified method on the half-line: II. NLS on the half-line with \(t\)-periodic boundary conditions, J. Phys. A Math. Theor., 45, 195202 (2012) · Zbl 1256.35045 · doi:10.1088/1751-8113/45/19/195202
[35] Hwang, G.; Fokas, AS, The modified Korteweg-de Vries equation on the half-line with a sine-wave as Dirichlet datum, J. Nonlinear Math. Phys., 20, 135-157 (2013) · Zbl 1420.37067 · doi:10.1080/14029251.2013.792492
[36] Lenells, J.; Fokas, AS, The nonlinear Schrödinger equation with \(t\)-periodic data: II. Perturbative results, Proc. R. Soc. A, 471, 20140926 (2015) · Zbl 1371.35274 · doi:10.1098/rspa.2014.0926
[37] Hwang, G., A perturbative approach for the asymptotic evaluation of the Neumann value corresponding to the Dirichlet datum of a single periodic exponential for the NLS, J. Nonlinear Math. Phys., 21, 225-247 (2014) · Zbl 1420.35359 · doi:10.1080/14029251.2014.905298
[38] Hwang, G., The modified Korteweg-de Vries equation on the quarter plane with \(t\)-periodic data, J. Nonlinear Math. Phys., 24, 620-634 (2017) · Zbl 1420.37099 · doi:10.1080/14029251.2017.1375695
[39] Colbrook, MJ; Fokas, AS; Hashemzadeh, P., A hybrid analytical-numerical technique for elliptic PDEs, SIAM J. Sci. Comput., 41, A1066-A1090 (2019) · Zbl 1414.65039 · doi:10.1137/18M1217309
[40] de Barros, FRJ; Colbrook, MJ; Fokas, AS, A hybrid analytical-numerical method for solving advection-dispersion problems on a half -line, Int. J. Heat Mass Transf., 139, 482-491 (2019) · doi:10.1016/j.ijheatmasstransfer.2019.05.018
[41] Hwang, G., Initial-boundary value problems for the one-dimensional linear advection-dispersion equation with decay, Z. Naturforshc. A, 75, 713-725 (2020) · doi:10.1515/zna-2020-0106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.