{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T10:19:14Z","timestamp":1772187554273,"version":"3.50.1"},"reference-count":66,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,2,25]],"date-time":"2020-02-25T00:00:00Z","timestamp":1582588800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,25]],"date-time":"2020-02-25T00:00:00Z","timestamp":1582588800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,5]]},"DOI":"10.1007\/s40314-020-1113-0","type":"journal-article","created":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T13:19:32Z","timestamp":1583500772000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["A meshfree approach for analysis and computational modeling of non-linear Schr\u00f6dinger equation"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6822-0566","authenticated-orcid":false,"given":"Ram","family":"Jiwari","sequence":"first","affiliation":[]},{"given":"Sanjay","family":"Kumar","sequence":"additional","affiliation":[]},{"given":"R. C.","family":"Mittal","sequence":"additional","affiliation":[]},{"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,25]]},"reference":[{"issue":"1","key":"1113_CR1","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/0045-7825(87)90117-4","volume":"61","author":"J Argyris","year":"1987","unstructured":"Argyris J, Haase M (1987) An engineer\u2019s guide to soliton phenomena: application of the finite element method. Comput Methods Appl Mech Eng 61(1):71\u2013122","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1113_CR2","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.apnum.2017.06.013","volume":"121","author":"A \u00c1vila","year":"2017","unstructured":"\u00c1vila A, Meister A, Steigemann M (2017) An adaptive galerkin method for the time-dependent complex Schr\u00f6dinger equation. Appl Numer Math 121:149\u2013169","journal-title":"Appl Numer Math"},{"issue":"7","key":"1113_CR3","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1016\/j.cpc.2007.05.010","volume":"177","author":"A Ayd\u0131n","year":"2007","unstructured":"Ayd\u0131n A, Karas\u00f6zen B (2007) Symplectic and multi-symplectic methods for coupled non-linear Schr\u00f6dinger equations with periodic solutions. Comput Phys Commun 177(7):566\u2013583","journal-title":"Comput Phys Commun"},{"key":"1113_CR4","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.chaos.2017.04.038","volume":"100","author":"A Bashan","year":"2017","unstructured":"Bashan A, Yagmurlu NM, Ucar Y, Esen A (2017) An effective approach to numerical soliton solutions for the Schr\u00f6dinger equation via modified cubic b-spline differential quadrature method. Chaos Solitons Fractals 100:45\u201356","journal-title":"Chaos Solitons Fractals"},{"issue":"2","key":"1113_CR5","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/0022-247X(71)90110-7","volume":"34","author":"R Bellman","year":"1971","unstructured":"Bellman R, Casti J (1971) Differential quadrature and long-term integration. J Math Anal Appl 34(2):235\u2013238","journal-title":"J Math Anal Appl"},{"issue":"10","key":"1113_CR6","doi-asserted-by":"crossref","first-page":"2026","DOI":"10.1016\/j.optcom.2010.01.046","volume":"283","author":"A Borhanifar","year":"2010","unstructured":"Borhanifar A, Abazari R (2010) Numerical study of non-linear Schr\u00f6dinger and coupled Schr\u00f6dinger equations by differential transformation method. Opt Commun 283(10):2026\u20132031","journal-title":"Opt Commun"},{"key":"1113_CR7","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511543241","volume-title":"Radial basis functions: theory and implementations","author":"MD Buhmann","year":"2003","unstructured":"Buhmann MD (2003) Radial basis functions: theory and implementations, vol 12. Cambridge University Press, Cambridge"},{"issue":"1\u20132","key":"1113_CR8","first-page":"247","volume":"174","author":"I Dag","year":"1999","unstructured":"Dag I (1999) A quadratic b-spline finite element method for solving non-linear schr\u00f6dinger equation. Comput Methods Appl Mech Eng 174(1\u20132):247\u2013258","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"1113_CR9","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.matcom.2005.10.001","volume":"71","author":"M Dehghan","year":"2006","unstructured":"Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71(1):16\u201330","journal-title":"Math Comput Simul"},{"key":"1113_CR10","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/j.jcp.2017.09.007","volume":"351","author":"M Dehghan","year":"2017","unstructured":"Dehghan M, Abbaszadeh M (2017) The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations. J Comput Phys 351:478\u2013510","journal-title":"J Comput Phys"},{"key":"1113_CR11","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.enganabound.2017.10.016","volume":"92","author":"M Dehghan","year":"2018","unstructured":"Dehghan M, Abbaszadeh M (2018) Solution of multi-dimensional Klein\u2013Gordon\u2013Zakharov and Schr\u00f6dinger\/gross\u2013Pitaevskii equations via local radial basis functions-differential quadrature (RBF-DQ) technique on non-rectangular computational domains. Eng Anal Bound Elem 92:156\u2013170","journal-title":"Eng Anal Bound Elem"},{"key":"1113_CR12","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.enganabound.2014.10.008","volume":"51","author":"M Dehghan","year":"2015","unstructured":"Dehghan M, Mohammadi V (2015) The numerical solution of Cahn\u2013Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFS) and RBFS-differential quadrature (RBFS-DQ) methods. Eng Anal Bound Elem 51:74\u2013100","journal-title":"Eng Anal Bound Elem"},{"key":"1113_CR13","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.cpc.2017.03.012","volume":"217","author":"M Dehghan","year":"2017","unstructured":"Dehghan M, Mohammadi V (2017) A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional non-linear Schr\u00f6dinger equations using an explicit time discretization: Runge-Kutta method. Comput Phys Commun 217:23\u201334","journal-title":"Comput Phys Commun"},{"issue":"18\u201319","key":"1113_CR14","doi-asserted-by":"crossref","first-page":"8578","DOI":"10.1016\/j.apm.2013.03.054","volume":"37","author":"M Dehghan","year":"2013","unstructured":"Dehghan M, Nikpour A (2013) Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Appl Math Model 37(18\u201319):8578\u20138599","journal-title":"Appl Math Model"},{"issue":"1","key":"1113_CR15","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.camwa.2007.01.038","volume":"54","author":"M Dehghan","year":"2007","unstructured":"Dehghan M, Shokri A (2007) A numerical method for two-dimensional Schr\u00f6dinger equation using collocation and radial basis functions. Comput Math Appl 54(1):136\u2013146","journal-title":"Comput Math Appl"},{"issue":"1","key":"1113_CR16","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.cpc.2009.08.015","volume":"181","author":"M Dehghan","year":"2010","unstructured":"Dehghan M, Taleei A (2010) A compact split-step finite difference method for solving the non-linear schr\u00f6dinger equations with constant and variable coefficients. Comput Phys Commun 181(1):43\u201351","journal-title":"Comput Phys Commun"},{"issue":"2","key":"1113_CR17","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0021-9991(81)90052-8","volume":"44","author":"M Delfour","year":"1981","unstructured":"Delfour M, Fortin M, Payr G (1981) Finite-difference solutions of a non-linear Schr\u00f6dinger equation. J Comput Phys 44(2):277\u2013288","journal-title":"J Comput Phys"},{"issue":"4","key":"1113_CR18","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1103\/PhysRevLett.71.533","volume":"71","author":"GC Duree Jr","year":"1993","unstructured":"Duree GC Jr, Shultz JL, Salamo GJ, Segev M, Yariv A, Crosignani B, Di Porto P, Sharp EJ, Neurgaonkar RR (1993) Observation of self-trapping of an optical beam due to the photorefractive effect. Phys Rev Lett 71(4):533","journal-title":"Phys Rev Lett"},{"key":"1113_CR19","doi-asserted-by":"crossref","DOI":"10.1142\/6437","volume-title":"Meshfree approximation methods with MATLAB","author":"GE Fasshauer","year":"2007","unstructured":"Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. World Scientific, Singapore"},{"key":"1113_CR20","doi-asserted-by":"crossref","unstructured":"Franke R (1979) A critical comparison of some methods for interpolation of scattered data. In: Technical Report, Naval Postgraduate School, Monterey, California","DOI":"10.21236\/ADA081688"},{"issue":"157","key":"1113_CR21","first-page":"181","volume":"38","author":"R Franke","year":"1982","unstructured":"Franke R (1982) Scattered data interpolation: tests of some methods. Math Comput 38(157):181\u2013200","journal-title":"Math Comput"},{"issue":"3\u20134","key":"1113_CR22","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/0045-7825(93)90007-K","volume":"108","author":"L Gardner","year":"1993","unstructured":"Gardner L, Gardner G, Zaki S, El Sahrawi Z (1993) B-spline finite element studies of the non-linear Schr\u00f6dinger equation. Comput Methods Appl Mech Eng 108(3\u20134):303\u2013318","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"3","key":"1113_CR23","doi-asserted-by":"crossref","first-page":"1587","DOI":"10.1007\/s11071-013-0887-7","volume":"73","author":"L Guo","year":"2013","unstructured":"Guo L, Guo Y, Billings S, Coca D, Lang Z (2013) The use of volterra series in the analysis of the non-linear Schr\u00f6dinger equation. Nonlinear Dyn 73(3):1587\u20131599","journal-title":"Nonlinear Dyn"},{"issue":"8","key":"1113_CR24","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1029\/JB076i008p01905","volume":"76","author":"RL Hardy","year":"1971","unstructured":"Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905\u20131915","journal-title":"J Geophys Res"},{"issue":"8\u20139","key":"1113_CR25","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/0898-1221(90)90272-L","volume":"19","author":"RL Hardy","year":"1990","unstructured":"Hardy RL (1990) Theory and applications of the multiquadric\u2013biharmonic method 20 years of discovery 1968\u20131988. Comput Math Appl 19(8\u20139):163\u2013208","journal-title":"Comput Math Appl"},{"issue":"3","key":"1113_CR26","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1063\/1.1654836","volume":"23","author":"A Hasegawa","year":"1973","unstructured":"Hasegawa A, Tappert F (1973) Transmission of stationary non-linear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl Phys Lett 23(3):142\u2013144","journal-title":"Appl Phys Lett"},{"key":"1113_CR27","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.apnum.2018.01.004","volume":"127","author":"J Hong","year":"2018","unstructured":"Hong J, Ji L, Liu Z (2018) Optimal error estimate of conservative local discontinuous galerkin method for non-linear schr\u00f6dinger equation. Appl Numer Math 127:164\u2013178","journal-title":"Appl Numer Math"},{"key":"1113_CR28","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.cpc.2014.11.004","volume":"188","author":"R Jiwari","year":"2015","unstructured":"Jiwari R (2015) A hybrid numerical scheme for the numerical solution of the burgers equation. Comput Phys Commun 188:59\u201367","journal-title":"Comput Phys Commun"},{"issue":"12","key":"1113_CR29","first-page":"6680","volume":"219","author":"R Jiwari","year":"2013","unstructured":"Jiwari R, Mittal R, Sharma KK (2013) A numerical scheme based on weighted average differential quadrature method for the numerical solution of burgers equation. Appl Math Comput 219(12):6680\u20136691","journal-title":"Appl Math Comput"},{"key":"1113_CR30","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1016\/j.chaos.2017.06.023","volume":"103","author":"R Jiwari","year":"2017","unstructured":"Jiwari R, Singh S, Kumar A (2017) Numerical simulation to capture the pattern formation of coupled reaction\u2013diffusion models. Chaos Solitons Fractals 103:422\u2013439","journal-title":"Chaos Solitons Fractals"},{"issue":"4","key":"1113_CR31","doi-asserted-by":"crossref","first-page":"1142","DOI":"10.1108\/EC-04-2018-0189","volume":"36","author":"R Jiwari","year":"2019","unstructured":"Jiwari R, Kumar S, Mittal R (2019) Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of burgers type problems. Eng Comput 36(4):1142\u20131168","journal-title":"Eng Comput"},{"issue":"8\u20139","key":"1113_CR32","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0898-1221(90)90271-K","volume":"19","author":"EJ Kansa","year":"1990","unstructured":"Kansa EJ (1990) Multiquadricsa scattered data approximation scheme with applications to computational fluid-dynamicsii solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19(8\u20139):147\u2013161","journal-title":"Comput Math Appl"},{"issue":"8\u20139","key":"1113_CR33","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/0898-1221(90)90270-T","volume":"19","author":"EJ Kansa","year":"1990","unstructured":"Kansa EJ (1990) Multiquadricsa scattered data approximation scheme with applications to computational fluid-dynamicsi surface approximations and partial derivative estimates. Comput Math Appl 19(8\u20139):127\u2013145","journal-title":"Comput Math Appl"},{"issue":"2","key":"1113_CR34","first-page":"215","volume":"6","author":"G Karamali","year":"2018","unstructured":"Karamali G, Abbaszadeh M, Dehghan M (2018) The smoothed particle hydrodynamics method for solving generalized variable coefficient Schr\u00f6dinger equation and Schr\u00f6dinger\u2013Boussinesq system. Comput Methods Differ Equ 6(2):215\u2013237","journal-title":"Comput Methods Differ Equ"},{"issue":"1","key":"1113_CR35","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1103\/RevModPhys.83.247","volume":"83","author":"YV Kartashov","year":"2011","unstructured":"Kartashov YV, Malomed BA, Torner L (2011) Solitons in nonlinear lattices. Rev Mod Phys 83(1):247","journal-title":"Rev Mod Phys"},{"key":"1113_CR36","volume-title":"Optical solitons: from fibers to photonic crystals","author":"YS Kivshar","year":"2003","unstructured":"Kivshar YS, Agrawal G (2003) Optical solitons: from fibers to photonic crystals. Academic Press, Cambridge"},{"issue":"6","key":"1113_CR37","doi-asserted-by":"crossref","first-page":"1814","DOI":"10.1002\/num.21969","volume":"31","author":"L Kong","year":"2015","unstructured":"Kong L, Hong J, Ji L, Zhu P (2015) Compact and efficient conservative schemes for coupled non-linear Schr\u00f6dinger equations. Numer Methods Partial Differ Equ 31(6):1814\u20131843","journal-title":"Numer Methods Partial Differ Equ"},{"issue":"1\u20132","key":"1113_CR38","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s11071-008-9380-0","volume":"56","author":"A Korkmaz","year":"2009","unstructured":"Korkmaz A, Da\u011f \u0130 (2009) A differential quadrature algorithm for non-linear Schr\u00f6dinger equation. Nonlinear Dyn 56(1\u20132):69\u201383","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"1113_CR39","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s10910-018-0941-2","volume":"57","author":"S Kumar","year":"2019","unstructured":"Kumar S, Jiwari R, Mittal R (2019) Numerical simulation for computational modelling of reaction\u2013diffusion brusselator model arising in chemical processes. J Math Chem 57(1):149\u2013179","journal-title":"J Math Chem"},{"issue":"6","key":"1113_CR40","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1016\/j.cpc.2014.02.023","volume":"185","author":"V Ledoux","year":"2014","unstructured":"Ledoux V, Van Daele M (2014) The accurate numerical solution of the Schr\u00f6dinger equation with an explicitly time-dependent hamiltonian. Comput Phys Commun 185(6):1589\u20131594","journal-title":"Comput Phys Commun"},{"issue":"6","key":"1113_CR41","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1080\/00207160.2016.1184264","volume":"94","author":"H Li","year":"2017","unstructured":"Li H, Wang Y (2017) An averaged vector field legendre spectral element method for the non-linear Schr\u00f6dinger equation. Int J Comput Math 94(6):1196\u20131218","journal-title":"Int J Comput Math"},{"key":"1113_CR42","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.jcp.2014.11.014","volume":"282","author":"W Lu","year":"2015","unstructured":"Lu W, Huang Y, Liu H (2015) Mass preserving discontinuous galerkin methods for Schr\u00f6dinger equations. J Comput Phys 282:210\u2013226","journal-title":"J Comput Phys"},{"issue":"8","key":"1113_CR43","first-page":"2835","volume":"215","author":"W-X Ma","year":"2009","unstructured":"Ma W-X, Chen M (2009) Direct search for exact solutions to the non-linear schr\u00f6dinger equation. Appl Math Comput 215(8):2835\u20132842","journal-title":"Appl Math Comput"},{"key":"1113_CR44","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.cam.2015.08.009","volume":"299","author":"JE Mac\u00edas-D\u00edaz","year":"2016","unstructured":"Mac\u00edas-D\u00edaz JE, Tomasiello S (2016) A differential quadrature-based approach a la picard for systems of partial differential equations associated with fuzzy differential equations. J Comput Appl Math 299:15\u201323","journal-title":"J Comput Appl Math"},{"issue":"11","key":"1113_CR45","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1119\/1.10583","volume":"44","author":"E MacKinnon","year":"1976","unstructured":"MacKinnon E (1976) De broglies thesis: a critical retrospective. Am J Phys 44(11):1047\u20131055","journal-title":"Am J Phys"},{"issue":"1","key":"1113_CR46","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1007\/s11071-012-0481-4","volume":"70","author":"G-Q Meng","year":"2012","unstructured":"Meng G-Q, Gao Y-T, Yu X, Shen Y-J, Qin Y (2012) Multi-soliton solutions for the coupled non-linear Schr\u00f6dinger-type equations. Nonlinear Dyn 70(1):609\u2013617","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"1113_CR47","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1108\/02644401311286071","volume":"30","author":"R Mittal","year":"2012","unstructured":"Mittal R, Jiwari R, Sharma KK (2012) A numerical scheme based on differential quadrature method to solve time dependent burgers equation. Eng Comput 30(1):117\u2013131","journal-title":"Eng Comput"},{"issue":"13","key":"1113_CR48","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1103\/PhysRevLett.45.1095","volume":"45","author":"LF Mollenauer","year":"1980","unstructured":"Mollenauer LF, Stolen RH, Gordon JP (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45(13):1095","journal-title":"Phys Rev Lett"},{"issue":"9","key":"1113_CR49","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/0375-9601(84)90827-2","volume":"100","author":"G Neugebauer","year":"1984","unstructured":"Neugebauer G, Meinel R (1984) General n-soliton solution of the akns class on arbitrary background. Phys Lett A 100(9):467\u2013470","journal-title":"Phys Lett A"},{"issue":"1\u20132","key":"1113_CR50","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s11071-006-9142-9","volume":"50","author":"BT Nohara","year":"2007","unstructured":"Nohara BT (2007) Governing equations of envelopes created by nearly bichromatic waves on deep water. Nonlinear Dyn 50(1\u20132):49\u201360","journal-title":"Nonlinear Dyn"},{"issue":"7","key":"1113_CR51","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0898-1221(97)00042-4","volume":"33","author":"M Robinson","year":"1997","unstructured":"Robinson M (1997) The solution of non-linear Schr\u00f6dinger equations using orthogonal spline collocation. Comput Math Appl 33(7):39\u201357","journal-title":"Comput Math Appl"},{"issue":"6","key":"1113_CR52","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1103\/PhysRev.28.1049","volume":"28","author":"E Schr\u00f6dinger","year":"1926","unstructured":"Schr\u00f6dinger E (1926) An undulatory theory of the mechanics of atoms and molecules. Phys Rev 28(6):1049","journal-title":"Phys Rev"},{"issue":"10","key":"1113_CR53","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1109\/PROC.1973.9296","volume":"61","author":"AC Scott","year":"1973","unstructured":"Scott AC, Chu F, McLaughlin DW (1973) The soliton: a new concept in applied science. Proc IEEE 61(10):1443\u20131483","journal-title":"Proc IEEE"},{"key":"1113_CR54","volume-title":"Differential equations with applications and historical notes","author":"GF Simmons","year":"2016","unstructured":"Simmons GF (2016) Differential equations with applications and historical notes. CRC Press, Boca Raton"},{"key":"1113_CR55","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.enganabound.2019.03.014","volume":"104","author":"Z Sun","year":"2019","unstructured":"Sun Z (2019) A meshless symplectic method for two-dimensional non-linear Schr\u00f6dinger equations based on radial basis function approximation. Eng Anal Bound Elem 104:1\u20137","journal-title":"Eng Anal Bound Elem"},{"key":"1113_CR56","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/0021-9991(84)90003-2","volume":"55","author":"TR Taha","year":"1984","unstructured":"Taha TR (1984) Mj ablowitz, analytical and numerical aspects of certain non-linear evolution equations II. Numerical, nonlinear Schr\u00f6dinger equation. J Comput Phys 55:203\u2013230","journal-title":"J Comput Phys"},{"issue":"8","key":"1113_CR57","doi-asserted-by":"crossref","first-page":"3183","DOI":"10.1016\/j.camwa.2011.08.031","volume":"62","author":"S Tomasiello","year":"2011","unstructured":"Tomasiello S (2011) A note on three numerical procedures to solve volterra integro-differential equations in structural analysis. Comput Math Appl 62(8):3183\u20133193","journal-title":"Comput Math Appl"},{"issue":"1","key":"1113_CR58","first-page":"399","volume":"219","author":"S Tomasiello","year":"2012","unstructured":"Tomasiello S (2012) Some remarks on a new dq-based method for solving a class of volterra integro-differential equations. Appl Math Comput 219(1):399\u2013407","journal-title":"Appl Math Comput"},{"key":"1113_CR59","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.apnum.2018.02.005","volume":"128","author":"B Wang","year":"2018","unstructured":"Wang B, Liang D (2018) The finite difference scheme for non-linear Schr\u00f6dinger equations on unbounded domain by artificial boundary conditions. Appl Numer Math 128:183\u2013204","journal-title":"Appl Numer Math"},{"issue":"8","key":"1113_CR60","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1080\/00207160.2017.1322692","volume":"95","author":"J Wang","year":"2018","unstructured":"Wang J, Wang Y (2018) Numerical analysis of a new conservative scheme for the coupled non-linear Schr\u00f6dinger equations. Int J Comput Math 95(8):1583\u20131608","journal-title":"Int J Comput Math"},{"issue":"1","key":"1113_CR61","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/BF02123482","volume":"4","author":"H Wendland","year":"1995","unstructured":"Wendland H (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4(1):389\u2013396","journal-title":"Adv Comput Math"},{"issue":"4","key":"1113_CR62","doi-asserted-by":"crossref","first-page":"2825","DOI":"10.1007\/s11071-018-4724-x","volume":"95","author":"OP Yadav","year":"2019","unstructured":"Yadav OP, Jiwari R (2019) Some soliton-type analytical solutions and numerical simulation of non-linear Schr\u00f6dinger equation. Nonlinear Dyn 95(4):2825\u20132836","journal-title":"Nonlinear Dyn"},{"key":"1113_CR63","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1016\/j.jcp.2013.09.016","volume":"256","author":"X Yang","year":"2014","unstructured":"Yang X, Zhang H, Xu D (2014) Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation. J Comput Phys 256:824\u2013837","journal-title":"J Comput Phys"},{"issue":"5","key":"1113_CR64","first-page":"823","volume":"37","author":"V Zakharov","year":"1973","unstructured":"Zakharov V, Shabat A (1973) Interaction between solitons in a stable medium. Sov Phys JETP 37(5):823\u2013828","journal-title":"Sov Phys JETP"},{"issue":"1","key":"1113_CR65","first-page":"118134","volume":"61","author":"V Zakharov","year":"1971","unstructured":"Zakharov V, Shabat A, Zakharov V, Shabat A (1971) Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media. ksper Teoret Fiz 61(1):118134","journal-title":"ksper Teoret Fiz"},{"key":"1113_CR66","first-page":"333","volume":"249","author":"L Zhang","year":"2014","unstructured":"Zhang L, Liew K-M (2014) An element-free based solution for non-linear Schr\u00f6dinger equations using the Icvmls\u2013Ritz method. Appl Math Comput 249:333\u2013345","journal-title":"Appl Math Comput"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1113-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-020-1113-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-1113-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,24]],"date-time":"2021-02-24T01:23:58Z","timestamp":1614129838000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-020-1113-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,25]]},"references-count":66,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,5]]}},"alternative-id":["1113"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-1113-0","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,25]]},"assertion":[{"value":"13 August 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 December 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 February 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 February 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"95"}}