Tsuchida, Takayuki; Ujino, Hideaki; Wadati, Miki Integrable semi-discretization of the coupled modified KdV equations. (English) Zbl 0933.35176 J. Math. Phys. 39, No. 9, 4785-4813 (1998). Summary: The discrete version of the inverse scattering method proposed by Ablowitz and Ladik is extended to solve multi-component systems \[ {\partial u^{(i)}_n\over \partial t}=\left(1+ \sum^{M-1}_{j,k=0} C_{jk} u_n^{(j)} u_n^{(k)}\right) \bigl(u^{(i)}_{u+1} -u_{n-1}^{(i)}\bigr), \quad i=0, 1,\dots, M-1. \] The extension enables one to solve the initial value problem, which proves directly the complete integrability of a semi-discrete version of the coupled modified Korteweg-de Vries equations and their hierarchy. It also provides a procedure to obtain conservation laws and multi-soliton solutions of the hierarchy. Cited in 35 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 39A12 Discrete version of topics in analysis Keywords:discrete version of the inverse scattering method; coupled modified Korteweg-de Vries equations; conservation laws; multi-soliton solutions Software:CONDENS PDFBibTeX XMLCite \textit{T. Tsuchida} et al., J. Math. Phys. 39, No. 9, 4785--4813 (1998; Zbl 0933.35176) Full Text: DOI References: [1] DOI: 10.1103/PhysRevLett.19.1095 · doi:10.1103/PhysRevLett.19.1095 [2] DOI: 10.1143/JPSJ.32.1681 · doi:10.1143/JPSJ.32.1681 [3] DOI: 10.1143/JPSJ.34.1289 · Zbl 1334.35299 · doi:10.1143/JPSJ.34.1289 [4] DOI: 10.1143/JPSJ.33.1456 · doi:10.1143/JPSJ.33.1456 [5] Zakharov V. E., Sov. Phys. JETP 34 pp 62– (1972) [6] DOI: 10.1103/PhysRevLett.31.125 · Zbl 1243.35143 · doi:10.1103/PhysRevLett.31.125 [7] DOI: 10.1002/sapm19765519 · doi:10.1002/sapm19765519 [8] Manakov S. V., Sov. Phys. JETP 38 pp 248– (1974) [9] DOI: 10.1143/JPSJ.60.409 · Zbl 0920.35128 · doi:10.1143/JPSJ.60.409 [10] DOI: 10.1016/0167-2789(82)90004-5 · Zbl 1194.35524 · doi:10.1016/0167-2789(82)90004-5 [11] DOI: 10.1088/0266-5611/5/2/002 · Zbl 0685.35080 · doi:10.1088/0266-5611/5/2/002 [12] DOI: 10.1143/JPSJ.67.1175 · Zbl 0973.35170 · doi:10.1143/JPSJ.67.1175 [13] DOI: 10.1007/BF02099265 · Zbl 0753.35100 · doi:10.1007/BF02099265 [14] DOI: 10.1143/JPSJ.66.577 · Zbl 0946.35078 · doi:10.1143/JPSJ.66.577 [15] DOI: 10.1143/PTP.51.703 · Zbl 0942.37505 · doi:10.1143/PTP.51.703 [16] Manakov S. V., Sov. Phys. JETP 40 pp 269– (1975) [17] DOI: 10.1063/1.522558 · Zbl 0296.34062 · doi:10.1063/1.522558 [18] DOI: 10.1063/1.523009 · Zbl 0322.42014 · doi:10.1063/1.523009 [19] DOI: 10.1143/JPSJ.66.1939 · Zbl 0965.35502 · doi:10.1143/JPSJ.66.1939 [20] M. Hisakado, solv-int/9701022. · Zbl 0965.35502 · doi:10.1143/JPSJ.66.1939 [21] DOI: 10.1143/JPSJ.66.2530 · Zbl 0944.35074 · doi:10.1143/JPSJ.66.2530 [22] DOI: 10.1143/PTP.53.419 · Zbl 1079.35506 · doi:10.1143/PTP.53.419 [23] DOI: 10.1143/PTP.52.397 · Zbl 1098.81714 · doi:10.1143/PTP.52.397 [24] DOI: 10.1063/1.1666399 · Zbl 0257.35052 · doi:10.1063/1.1666399 [25] DOI: 10.1143/PTPS.59.36 · doi:10.1143/PTPS.59.36 [26] DOI: 10.1016/0370-2693(79)90079-0 · doi:10.1016/0370-2693(79)90079-0 [27] DOI: 10.1016/S0375-9601(97)00750-0 · Zbl 0969.35542 · doi:10.1016/S0375-9601(97)00750-0 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.