{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T08:31:23Z","timestamp":1764059483219,"version":"3.37.3"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T00:00:00Z","timestamp":1704326400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T00:00:00Z","timestamp":1704326400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12226313 and 12226319"],"award-info":[{"award-number":["12226313 and 12226319"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2024,11]]},"DOI":"10.1007\/s11075-023-01739-y","type":"journal-article","created":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T11:01:56Z","timestamp":1704366116000},"page":"1017-1037","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Artificial boundary method for the Zakharov-Rubenchik equations"],"prefix":"10.1007","volume":"97","author":[{"given":"Hongwei","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiangyu","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,1,4]]},"reference":[{"key":"1739_CR1","first-page":"84","volume":"5","author":"VE Zakharov","year":"1972","unstructured":"Zakharov, V.E., Rubenchik, A.M.: Nonlinear interaction between high and low frequency waves. Prikl. Mat. Techn. Fiz. 5, 84\u201389 (1972)","journal-title":"Prikl. Mat. Techn. Fiz."},{"key":"1739_CR2","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1016\/S0167-2789(02)00722-4","volume":"175","author":"F Oliveira","year":"2003","unstructured":"Oliveira, F.: Stability of the solitons for the one-dimensional Zakharov-Rubenchik equation. Phys. D 175, 220\u2013240 (2003)","journal-title":"Phys. D"},{"key":"1739_CR3","doi-asserted-by":"publisher","first-page":"693","DOI":"10.4310\/PAMQ.2017.v13.n4.a4","volume":"13","author":"X Liu","year":"2017","unstructured":"Liu, X.: Orbital stability of solitary wave solutions of Zakharov-Rubenchik equation. Pure Appl. Math. Q. 13, 693\u2013710 (2017)","journal-title":"Pure Appl. Math. Q."},{"key":"1739_CR4","doi-asserted-by":"publisher","first-page":"663","DOI":"10.22199\/issn.0717-6279-4547","volume":"41","author":"JR Quintero","year":"2022","unstructured":"Quintero, J.R.: Stability and instability analysis for the standing waves for a generalized Zakharov-Rubenchik system. Proyecciones 41, 663\u2013682 (2022)","journal-title":"Proyecciones"},{"key":"1739_CR5","first-page":"2927","volume":"13","author":"L Zhang","year":"2020","unstructured":"Zhang, L., Yuan, P., Fu, J., Khalique, C.M.: Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation. Discrete Contin. Dyn. Syst. Ser. S 13, 2927\u20132939 (2020)","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"1739_CR6","doi-asserted-by":"publisher","first-page":"412","DOI":"10.1007\/s10915-013-9768-y","volume":"59","author":"X Zhao","year":"2014","unstructured":"Zhao, X., Li, Z.: Numerical methods and simulations for the dynamics of one-dimensional Zakharov-Rubenchik equations. J. Sci. Comput. 59, 412\u2013438 (2014)","journal-title":"J. Sci. Comput."},{"key":"1739_CR7","doi-asserted-by":"publisher","first-page":"1163","DOI":"10.1007\/s10444-018-9651-3","volume":"45","author":"X Zhou","year":"2019","unstructured":"Zhou, X., Wang, T., Zhang, L.: Two numerical methods for the Zakharov-Rubenchik equations. Adv. Comput. Math. 45, 1163\u20131184 (2019)","journal-title":"Adv. Comput. Math."},{"key":"1739_CR8","doi-asserted-by":"publisher","first-page":"A929","DOI":"10.1137\/17M1113333","volume":"40","author":"W Bao","year":"2018","unstructured":"Bao, W., Su, C.: A uniformly and optimally accurate method for the Zakharov system in the subsonic limit regime. SIAM J. Sci. Comput. 40, A929\u2013A953 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1739_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.110064","volume":"428","author":"C Su","year":"2021","unstructured":"Su, C., Zhao, X.: A uniformly and optimally accurate method for the Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime. J. Comput. Phys. 428, 110064 (2021)","journal-title":"J. Comput. Phys."},{"key":"1739_CR10","doi-asserted-by":"publisher","first-page":"537","DOI":"10.1080\/00207160.2018.1437261","volume":"96","author":"B Ji","year":"2019","unstructured":"Ji, B., Zhang, L., Zhou, X.: Conservative compact difference scheme for the Zakharov-Rubenchik equations. Int. J. Comput. Math. 96, 537\u2013556 (2019)","journal-title":"Int. J. Comput. Math."},{"key":"1739_CR11","volume":"394","author":"\u00d6 Oruc","year":"2021","unstructured":"Oruc, \u00d6.: A radial basis function finite difference (RBF-FD) method for numerical simulation of interaction of high and low frequency waves: Zakharov-Rubenchik equations. Appl. Math. Comput. 394, 125787 (2021)","journal-title":"Appl. Math. Comput."},{"key":"1739_CR12","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/j.enganabound.2022.09.004","volume":"145","author":"M Dehghan","year":"2022","unstructured":"Dehghan, M., Hooshyarfarzin, B., Abbaszadeh, M.: Radial basis function partition of unity procedure combined with the reduced-order method for solving Zakharov-Rubenchik equations. Eng. Anal. Bound. Elem. 145, 93\u2013116 (2022)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1739_CR13","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1007\/s10915-022-02075-4","volume":"94","author":"G Zhang","year":"2023","unstructured":"Zhang, G., Jiang, C., Huang, H.: Arbitrarily high-order energy-preserving schemes for the Zakharov-Rubenchik equations. J. Sci. Comput. 94, 32 (2023)","journal-title":"J. Sci. Comput."},{"key":"1739_CR14","first-page":"729","volume":"4","author":"X Antoine","year":"2008","unstructured":"Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Sch\u00e4dle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schr\u00f6dinger equations. Commun. Comput. Phys. 4, 729\u2013796 (2008)","journal-title":"Commun. Comput. Phys."},{"key":"1739_CR15","doi-asserted-by":"publisher","first-page":"1577","DOI":"10.1016\/j.jcp.2007.02.004","volume":"225","author":"Z Xu","year":"2007","unstructured":"Xu, Z., Han, H., Wu, X.: Adaptive absorbing boundary conditions for Schr\u00f6dinger-type equations: application to nonlinear and multi-dimensional problems. J. Comput. Phys. 225, 1577\u20131589 (2007)","journal-title":"J. Comput. Phys."},{"key":"1739_CR16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.78.026709","volume":"78","author":"J Zhang","year":"2008","unstructured":"Zhang, J., Xu, Z., Wu, X.: Unified approach to split absorbing boundary conditions for nonlinear Schr\u00f6dinger equations. Phys. Rev. E 78, 026709 (2008)","journal-title":"Phys. Rev. E"},{"key":"1739_CR17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.79.046711","volume":"79","author":"J Zhang","year":"2009","unstructured":"Zhang, J., Xu, Z., Wu, X.: Unified approach to split absorbing boundary conditions for nonlinear Schr\u00f6dinger equations: two-dimensional case. Phys. Rev. E 79, 046711 (2009)","journal-title":"Phys. Rev. E"},{"key":"1739_CR18","doi-asserted-by":"publisher","first-page":"955","DOI":"10.1016\/S0898-1221(04)90079-X","volume":"47","author":"S Jiang","year":"2004","unstructured":"Jiang, S., Greengard, L.: Fast evaluation of nonreflecting boundary conditions for the Schr\u00f6dinger equation in one dimension. Comput. Math. Appl. 47, 955\u2013966 (2004)","journal-title":"Comput. Math. Appl."},{"key":"1739_CR19","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1002\/cpa.20200","volume":"61","author":"S Jiang","year":"2008","unstructured":"Jiang, S., Greengard, L.: Efficient representation of nonreflecting boundary conditions for the time-dependent Schr\u00f6dinger equation in two dimensions. Comm. Pure Appl. Math. 61, 261\u2013288 (2008)","journal-title":"Comm. Pure Appl. Math."},{"key":"1739_CR20","doi-asserted-by":"publisher","first-page":"A4083","DOI":"10.1137\/17M1162111","volume":"40","author":"B Li","year":"2018","unstructured":"Li, B., Zhang, J., Zheng, C.: Stability and error analysis for a second-order fast approximation of the one-dimensional Schr\u00f6dinger equation under absorbing boundary conditions. SIAM J. Sci. Comput. 40, A4083\u2013A4104 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1739_CR21","doi-asserted-by":"publisher","first-page":"766","DOI":"10.1137\/17M1122347","volume":"56","author":"B Li","year":"2018","unstructured":"Li, B., Zhang, J., Zheng, C.: An efficient second-order finite difference method for the one-dimensional Schr\u00f6dinger equation with absorbing boundary conditions. SIAM J. Numer. Anal. 56, 766\u2013791 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1739_CR22","doi-asserted-by":"publisher","first-page":"812","DOI":"10.1137\/0909054","volume":"9","author":"B Gustafsson","year":"1988","unstructured":"Gustafsson, B.: Far-field boundary conditions for time-dependent hyperbolic systems. SIAM J. Sci. Statist. Comput. 9, 812\u2013828 (1988)","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"1739_CR23","doi-asserted-by":"publisher","first-page":"295","DOI":"10.4208\/nmtma.2010.33.3","volume":"3","author":"M Ehrhardt","year":"2010","unstructured":"Ehrhardt, M.: Absorbing boundary conditions for hyperbolic systems. Numer. Math. Theory Methods Appl. 3, 295\u2013337 (2010)","journal-title":"Numer. Math. Theory Methods Appl."},{"key":"1739_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0021-9991(87)90041-6","volume":"68","author":"KW Thompson","year":"1987","unstructured":"Thompson, K.W.: Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68, 1\u201324 (1987)","journal-title":"J. Comput. Phys."},{"key":"1739_CR25","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/j.wavemoti.2003.12.004","volume":"39","author":"D Givoli","year":"2004","unstructured":"Givoli, D.: High-order local non-reflecting boundary conditions: a review. Wave Motion 39, 319\u2013326 (2004)","journal-title":"Wave Motion"},{"key":"1739_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.wavemoti.2022.103109","volume":"118","author":"E Turkel","year":"2023","unstructured":"Turkel, E., Gordon, R., Gordon, D.: Local absorbing boundary conditions for the elastic wave equation. Wave Motion 118, 103109 (2023)","journal-title":"Wave Motion"},{"key":"1739_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111351","volume":"466","author":"G Pang","year":"2022","unstructured":"Pang, G., Ji, S., Antoine, X.: Accurate absorbing boundary conditions for two-dimensional peridynamics. J. Comput. Phys. 466, 111351 (2022)","journal-title":"J. Comput. Phys."},{"key":"1739_CR28","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1016\/j.apnum.2020.04.008","volume":"156","author":"K Pan","year":"2020","unstructured":"Pan, K., Xia, J., He, D., Zhang, Q.: A three-level linearized difference scheme for nonlinear Schr\u00f6dinger equation with absorbing boundary conditions. Appl. Numer. Math. 156, 32\u201349 (2020)","journal-title":"Appl. Numer. Math."},{"key":"1739_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.109029","volume":"401","author":"A Modave","year":"2020","unstructured":"Modave, A., Geuzaine, C., Antoine, X.: Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering. J. Comput. Phys. 401, 109029 (2020)","journal-title":"J. Comput. Phys."},{"key":"1739_CR30","doi-asserted-by":"publisher","first-page":"1061","DOI":"10.1093\/imanum\/drab106","volume":"43","author":"T Sun","year":"2023","unstructured":"Sun, T., Zheng, C.: Efficient implementation of the exact artificial boundary condition for the exterior problem of the Stokes system in three dimensions. IMA J. Numer. Anal. 43, 1061\u20131088 (2023)","journal-title":"IMA J. Numer. Anal."},{"key":"1739_CR31","volume-title":"Pseudodifferential operators and spectral theory","author":"A Shubin","year":"1986","unstructured":"Shubin, A.: Pseudodifferential operators and spectral theory. Springer-Verlag, Berlin, New York (1986)"},{"key":"1739_CR32","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/0021-9991(90)90152-Q","volume":"89","author":"KW Thompson","year":"1990","unstructured":"Thompson, K.W.: Time-dependent boundary conditions for hyperbolic systems. II. J. Comput. Phys. 89, 439\u2013461 (1990)","journal-title":"II. J. Comput. Phys."},{"key":"1739_CR33","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1090\/S0025-5718-1986-0856695-2","volume":"176","author":"LN Trefethen","year":"1986","unstructured":"Trefethen, L.N., Halpern, L.: Well-posedness of one-way wave equations and absorbing boundary conditions. Math. Comput. 176, 421\u2013435 (1986)","journal-title":"Math. Comput."},{"key":"1739_CR34","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1016\/j.mcm.2010.09.005","volume":"53","author":"H Jia","year":"2011","unstructured":"Jia, H., Li, K.: A third accurate operator splitting method. Math. Comput. Model. 53, 387\u2013396 (2011)","journal-title":"Math. Comput. Model."},{"key":"1739_CR35","volume-title":"Lie groups","author":"VS Varadarajan\u00f3","year":"1974","unstructured":"Varadarajan\u00f3, V.S.: Lie groups. Lie Algebras and their Representation. Prentice-Hall, Englewood Cliffs, NJ (1974)"},{"key":"1739_CR36","doi-asserted-by":"publisher","first-page":"1527","DOI":"10.1137\/S0036142902418345","volume":"42","author":"J Szeftel","year":"2004","unstructured":"Szeftel, J.: Design of absorbing boundary conditions for Schr\u00f6dinger equations in $$\\mathbb{R} ^d$$. SIAM J. Numer. Anal. 42, 1527\u20131551 (2004)","journal-title":"SIAM J. Numer. Anal."},{"key":"1739_CR37","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s00211-006-0012-7","volume":"104","author":"J Szeftel","year":"2006","unstructured":"Szeftel, J.: Absorbing boundary conditions for one-dimensional nonlinear Schr\u00f6dinger equations. Numer. Math. 104, 103\u2013127 (2006)","journal-title":"Numer. Math."},{"key":"1739_CR38","doi-asserted-by":"publisher","first-page":"742","DOI":"10.4208\/cicp.280610.161110a","volume":"10","author":"J Zhang","year":"2011","unstructured":"Zhang, J., Sun, Z., Wu, X., Wang, D.: Analysis of high-order absorbing boundary conditions for the Schr\u00f6dinger equation. Commun. Comput. Phys. 10, 742\u2013766 (2011)","journal-title":"Commun. Comput. Phys."},{"key":"1739_CR39","doi-asserted-by":"publisher","first-page":"2030","DOI":"10.1121\/1.405255","volume":"92","author":"J Papadakis","year":"1992","unstructured":"Papadakis, J., Taroudakis, M., Papadakis, P., Mayfield, B.: A new method for a realistic treatment of the sea bottom in the parabolic approximation. J. Acoust. Soc. Am. 92, 2030\u20132038 (1992)","journal-title":"J. Acoust. Soc. Am."},{"key":"1739_CR40","unstructured":"Mayfield, M.E.: Non-reflective boundary conditions for Schroedinger\u2019s equation. Ph. D. thesis, University of Rhode Island (1989)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01739-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01739-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01739-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T10:09:08Z","timestamp":1728295748000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01739-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,4]]},"references-count":40,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["1739"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01739-y","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2024,1,4]]},"assertion":[{"value":"17 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 December 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 January 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}},{"value":"The authors consent to participate.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"The authors consent to publish this manuscript.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare no competing interests.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}