{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T11:20:04Z","timestamp":1784028004936,"version":"3.55.0"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2007,7,28]],"date-time":"2007-07-28T00:00:00Z","timestamp":1185580800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2007,12]]},"DOI":"10.1007\/s00332-007-9004-8","type":"journal-article","created":{"date-parts":[[2007,7,27]],"date-time":"2007-07-27T21:47:32Z","timestamp":1185572852000},"page":"569-607","source":"Crossref","is-referenced-by-count":30,"title":["A Numerical Study of the Stability of Solitary Waves of\u00a0the\u00a0Bona\u2013Smith Family of Boussinesq Systems"],"prefix":"10.1007","volume":"17","author":[{"given":"V. A.","family":"Dougalis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Dur\u00e1n","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M. A.","family":"L\u00f3pez-Marcos","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D. E.","family":"Mitsotakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2007,7,28]]},"reference":[{"key":"9004_CR1","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0167-2789(87)90084-4","volume":"24","author":"J. P. Albert","year":"1987","unstructured":"J. P. Albert, J. L. Bona, and D. Henry. Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Physica D, 24:343\u2013366, 1987.","journal-title":"Physica D"},{"key":"9004_CR2","unstructured":"D. C. Antonopoulos. The Boussinesq system of equations: Theory and numerical analysis. Ph.D. thesis. University of Athens, 2000 (in Greek)."},{"key":"9004_CR3","unstructured":"D. C. Antonopoulos and V. A. Dougalis. Numerical approximation of Boussinesq systems. In A.\u00a0Bermudez et\u00a0al., editors, Proceedings of the 5th International Conference on Mathematical and Numerical Aspects of Wave Propagation, pages 265\u2013269. SIAM, Philadelphia, 2000."},{"key":"9004_CR4","unstructured":"D. C. Antonopoulos, V. A. Dougalis, and D.\u00a0E. Mitsotakis. Theory and numerical analysis of the Bona\u2013Smith type systems of Boussinesq equations (to appear)."},{"key":"9004_CR5","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1098\/rspa.1972.0074","volume":"328","author":"T. B. Benjamin","year":"1972","unstructured":"T. B. Benjamin. The stability of solitary waves. Proc. R. Soc. Lond. Ser. A, 328:153\u2013183, 1972.","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"9004_CR6","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1098\/rspa.1975.0106","volume":"344","author":"J. L. Bona","year":"1975","unstructured":"J. L. Bona. On the stability theory of solitary waves. Proc. R. Soc. Lond. Ser. A, 344:363\u2013374, 1975.","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"9004_CR7","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0167-2789(97)00249-2","volume":"116","author":"J. L. Bona","year":"1998","unstructured":"J. L. Bona and M. Chen. A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D, 116:191\u2013224, 1998.","journal-title":"Physica D"},{"key":"9004_CR8","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s00332-002-0466-4","volume":"12","author":"J. L. Bona","year":"2002","unstructured":"J. L. Bona, M. Chen, and J.-C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and linear theory. J. Nonlinear Sci., 12:283\u2013318, 2002.","journal-title":"J. Nonlinear Sci."},{"key":"9004_CR9","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1088\/0951-7715\/17\/3\/010","volume":"17","author":"J. L. Bona","year":"2004","unstructured":"J. L. Bona, M. Chen, and J.-C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory. Nonlinearity, 17:925\u2013952, 2004.","journal-title":"Nonlinearity"},{"key":"9004_CR10","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1098\/rsta.1995.0027","volume":"351","author":"J. L. Bona","year":"1995","unstructured":"J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney. Conservative, high-order numerical schemes for the generalized Korteweg\u2013de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A, 351:107\u2013164, 1995.","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"9004_CR11","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1007\/s003320010003","volume":"10","author":"J. L. Bona","year":"2000","unstructured":"J. L. Bona, W. R. McKinney, and J. M. Restrepo. Unstable solitary-wave solutions of the generalized regularized long-wave equation. J. Nonlinear Sci., 10:603\u2013638, 2000.","journal-title":"J. Nonlinear Sci."},{"key":"9004_CR12","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1080\/03091928908219524","volume":"48","author":"J. L. Bona","year":"2000","unstructured":"J. L. Bona and R. L. Sachs. The existence of internal solitary waves in a two-fluid system near the KdV limit. Geophys. Astrophys. Fluid Dyn., 48:25\u201351, 2000.","journal-title":"Geophys. Astrophys. Fluid Dyn."},{"key":"9004_CR13","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1017\/S030500410005218X","volume":"79","author":"J. L. Bona","year":"1976","unstructured":"J. L. Bona and R. Smith. A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Philos. Soc., 79:167\u2013182, 1976.","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"9004_CR14","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1098\/rspa.1987.0073","volume":"411","author":"J. L. Bona","year":"1987","unstructured":"J. L. Bona, P. E. Souganidis, and W. A. Strauss. Stability and instability of solitary waves of KdV type. Proc. R. Soc. Lond. Ser. A, 411:395\u2013412, 1987.","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"9004_CR15","first-page":"55","volume":"17","author":"J. V. Boussinesq","year":"1872","unstructured":"J. V. Boussinesq. Th\u00e9orie des ondes et des remous qui se propagent le long d\u2019 un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pure Appl., 17:55\u2013108, 1872.","journal-title":"J. Math. Pure Appl."},{"key":"9004_CR16","first-page":"1","volume":"23","author":"J. V. Boussinesq","year":"1877","unstructured":"J. V. Boussinesq. Essai sur la th\u00e9orie des eaux courants. M\u00e9m. pr\u00e9s. div. sav. Acad. des Sci. Inst. Fr. (s\u00e9r. 2), 23:1\u2013680, 1877.","journal-title":"M\u00e9m. pr\u00e9s. div. sav. Acad. des Sci. Inst. Fr. (s\u00e9r. 2)"},{"key":"9004_CR17","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1023\/A:1026667903256","volume":"37","author":"M. Chen","year":"1998","unstructured":"M. Chen. Exact traveling-wave solutions to bi-directional wave equations. Int. J. Theor. Phys., 37:1547\u20131567, 1998.","journal-title":"Int. J. Theor. Phys."},{"key":"9004_CR18","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1080\/00036810008840844","volume":"75","author":"M. Chen","year":"2000","unstructured":"M. Chen. Solitary-wave and multipulsed traveling-wave solutions of Boussinesq systems. Appl. Anal., 75:213\u2013240, 2000.","journal-title":"Appl. Anal."},{"key":"9004_CR19","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1142\/9789812702593_0030","volume-title":"Advances in Scattering Theory and Biomedical Engineering","author":"V. A. Dougalis","year":"2004","unstructured":"V. A. Dougalis and D. E. Mitsotakis. Solitary waves of the Bona\u2013Smith system. In D.\u00a0Fotiadis and C.\u00a0Massalas, editors, Advances in Scattering Theory and Biomedical Engineering, pages 286\u2013294. World Scientific, River Edge, 2004."},{"key":"9004_CR20","doi-asserted-by":"crossref","first-page":"7761","DOI":"10.1088\/0305-4470\/36\/28\/306","volume":"36","author":"A. Dur\u00e1n","year":"2003","unstructured":"A. Dur\u00e1n and M. A. L\u00f3pez-Marcos. Conservative numerical methods for solitary wave interactions. J. Phys. A: Math. Gen., 36:7761\u20137770, 2003.","journal-title":"J. Phys. A: Math. Gen."},{"key":"9004_CR21","doi-asserted-by":"crossref","first-page":"583","DOI":"10.3934\/dcds.2005.13.583","volume":"13","author":"K. El Dika","year":"2005","unstructured":"K. El Dika. Asymptotic stability of solitary waves for the Benjamin\u2013Bona\u2013Mahony equation. Discrete. Contin. Dyn. Syst., 13:583\u2013622, 2005.","journal-title":"Discrete. Contin. Dyn. Syst."},{"key":"9004_CR22","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/0022-1236(87)90044-9","volume":"74","author":"M. Grillakis","year":"1987","unstructured":"M. Grillakis, J. Shatah, and W. A. Strauss. Stability of solitary waves in the presence of symmetry:\u00a0I. J. Funct. Anal., 74:170\u2013197, 1987.","journal-title":"J. Funct. Anal."},{"key":"9004_CR23","volume-title":"Perturbation Theory for Linear Operators","author":"T. Kato","year":"1980","unstructured":"T. Kato. Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin, 1980.","edition":"2"},{"key":"9004_CR24","doi-asserted-by":"crossref","first-page":"99","DOI":"10.2991\/jnmp.2002.9.s1.9","volume":"9","author":"Yi A. Li","year":"2002","unstructured":"Yi A. Li. Hamiltonian structure and linear stability of solitary waves of the Green\u2013Naghdi equation. J. Nonlinear Math. Phys., 9:99\u2013105, 2002. Suppl. I.","journal-title":"J. Nonlinear Math. Phys."},{"key":"9004_CR25","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s002050100138","volume":"157","author":"Y. Martel","year":"2001","unstructured":"Y. Martel and F. Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rat. Mech. Anal., 157:219\u2013254, 2001.","journal-title":"Arch. Rat. Mech. Anal."},{"key":"9004_CR26","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1002\/(SICI)1097-0312(199604)49:4<399::AID-CPA4>3.0.CO;2-7","volume":"49","author":"J. R. Miller","year":"1996","unstructured":"J. R. Miller and M. I. Weinstein. Asymptotic stability of solitary waves for the Regularized Long-Wave equation. Commun. Pure Appl. Math., 49:399\u2013441, 1996.","journal-title":"Commun. Pure Appl. Math."},{"key":"9004_CR27","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1088\/0951-7715\/8\/6\/003","volume":"8","author":"R. L. Pego","year":"1995","unstructured":"R. L. Pego, P. Smereka, and M. I. Weinstein. Oscillatory instability of solitary waves in a continuum model of lattice vibrations. Nonlinearity, 8:921\u2013941, 1995.","journal-title":"Nonlinearity"},{"key":"9004_CR28","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/BF02101705","volume":"164","author":"R. L. Pego","year":"1994","unstructured":"R. L. Pego and M. I. Weinstein. Asymptotic stability of solitary waves. Commun. Math. Phys., 164:305\u2013349, 1994.","journal-title":"Commun. Math. Phys."},{"key":"9004_CR29","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1111\/1467-9590.00063","volume":"99","author":"R. L. Pego","year":"1997","unstructured":"R. L. Pego and M. I. Weinstein. Convective linear stability of solitary waves for Boussinesq equations. Stud. Appl. Math., 99:311\u2013375, 1997.","journal-title":"Stud. Appl. Math."},{"key":"9004_CR30","unstructured":"B. Pelloni. Spectral methods for the numerical solution of nonlinear dispersive wave equations. Ph.D. thesis, Yale University, 1996."},{"key":"9004_CR31","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1016\/S0378-4754(00)00305-0","volume":"55","author":"B. Pelloni","year":"2001","unstructured":"B. Pelloni and V. A. Dougalis. Numerical modelling of two-way propagation of nonlinear dispersive waves. Math. Comput. Simul., 55:595\u2013606, 2001.","journal-title":"Math. Comput. Simul."},{"key":"9004_CR32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s003329910001","volume":"10","author":"B. Pelloni","year":"2000","unstructured":"B. Pelloni and V. A. Dougalis. Numerical solutions of some nonlocal, nonlinear, dispersive wave equations. J. Nonlin. Sci., 10:1\u201322, 2000.","journal-title":"J. Nonlin. Sci."},{"key":"9004_CR33","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/B978-0-12-493250-0.50007-2","volume-title":"Waves on Beaches and Resulting Sediment Transport","author":"D. H. Peregrine","year":"1972","unstructured":"D. H. Peregrine. Equations for water waves and the approximations behind them. In R. E. Meyer, editor, Waves on Beaches and Resulting Sediment Transport, pages 95\u2013121. Academic Press, New York, 1972."},{"key":"9004_CR34","volume-title":"Analysis of Operators IV","author":"M. Reed","year":"1978","unstructured":"M. Reed and B. Simon. Analysis of Operators IV. Academic Press, New York, 1978."},{"key":"9004_CR35","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0073054","volume-title":"Asymptotic Analysis of Soliton Problems: An Inverse Scattering Approach","author":"P. C. Schuur","year":"1986","unstructured":"P. C. Schuur. Asymptotic Analysis of Soliton Problems: An Inverse Scattering Approach, volume 1232 of Lecture Notes in Mathematics. Springer, Berlin, 1986."},{"key":"9004_CR36","first-page":"255","volume-title":"Nonlinear Dispersive Wave Systems","author":"P. Smereka","year":"1992","unstructured":"P. Smereka. A remark on the solitary wave stability for a Boussinesq equation. In L. Debnath, editor, Nonlinear Dispersive Wave Systems, pages 255\u2013263. World Scientific, Singapore, 1992."},{"key":"9004_CR37","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1017\/S0305004100058801","volume":"90","author":"J. F. Toland","year":"1981","unstructured":"J. F. Toland. Solitary wave solutions for a model of the two-way propagation of water waves in a channel. Math. Proc. Camb. Philos. Soc., 90:343\u2013360, 1981.","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"9004_CR38","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/BF01209303","volume":"94","author":"J. F. Toland","year":"1984","unstructured":"J. F. Toland. Uniqueness and a priori bounds for certain homoclinic orbits of a Boussinesq system modelling solitary water waves. Commun. Math. Phys., 94:239\u2013254, 1984.","journal-title":"Commun. Math. Phys."},{"key":"9004_CR39","first-page":"447","volume-title":"Proceedings of Symposia in Pure Mathematics, volume 45, Part 2","author":"J. F. Toland","year":"1986","unstructured":"J. F. Toland: Existence of symmetric homoclinic orbits for systems of Euler\u2013Lagrange equations. In Proceedings of Symposia in Pure Mathematics, volume 45, Part 2, pages 447\u2013459. Am. Math. Soc., Providence, 1986."},{"key":"9004_CR40","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1002\/cpa.3160390103","volume":"39","author":"M. Weinstein","year":"1986","unstructured":"M. Weinstein. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math., 39:51\u201368, 1986.","journal-title":"Commun. Pure Appl. Math."},{"key":"9004_CR41","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1080\/03605308708820522","volume":"12","author":"M. Weinstein","year":"1987","unstructured":"M. Weinstein. Existence and dynamic stability of solitary-wave solutions of equations arising in long wave propagation. Commun. Partial Differ. Eq., 12:1133\u20131173, 1987.","journal-title":"Commun. Partial Differ. Eq."},{"key":"9004_CR42","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1090\/conm\/200\/02517","volume":"200","author":"M. I. Weinstein","year":"1996","unstructured":"M. I. Weinstein. Asymptotic stability of nonlinear bound states in conservative dispersive systems. Contemp. Math., 200:223\u2013235, 1996.","journal-title":"Contemp. Math."},{"key":"9004_CR43","volume-title":"Linear and Non-linear Waves","author":"G. B. Whitham","year":"1974","unstructured":"G. B. Whitham. Linear and Non-linear Waves. Wiley, New York, 1974."},{"key":"9004_CR44","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1137\/0719037","volume":"19","author":"R. Winther","year":"1982","unstructured":"R. Winther. A finite element method for a version of the Boussinesq equations. SIAM J. Numer. Anal., 19:561\u2013570, 1982.","journal-title":"SIAM J. Numer. Anal."},{"key":"9004_CR45","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1103\/PhysRevLett.15.240","volume":"15","author":"N. J. Zabusky","year":"1965","unstructured":"N. J. Zabusky and M. D. Kruskal. Interaction of \u201csolitons\u201d in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15:240\u2013243, 1965.","journal-title":"Phys. Rev. Lett."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-007-9004-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-007-9004-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-007-9004-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T12:08:50Z","timestamp":1559045330000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-007-9004-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,7,28]]},"references-count":45,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2007,12]]}},"alternative-id":["9004"],"URL":"https:\/\/doi.org\/10.1007\/s00332-007-9004-8","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,7,28]]}}}