{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T06:46:18Z","timestamp":1772779578678,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Heilong Province of China","award":["PL2024A002"],"award-info":[{"award-number":["PL2024A002"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12571525"],"award-info":[{"award-number":["12571525"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Zhejiang Province of China","award":["LQ21A010007"],"award-info":[{"award-number":["LQ21A010007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates the conservation of mass and energy in the space-fractional Klein-Gordon-Schr\u00f6dinger system with fractional Laplacian operators. Firstly, the invariant energy quadratization method is applied to transform the original system into an equivalent form. For spatial discretization, Fourier spectral methods are employed, yielding a semi-discrete scheme. Subsequently, an invariant energy quadratization Runge-Kutta approach is used for temporal discretization, resulting in a fully discrete scheme. Owing to its diagonally implicit structure, the proposed scheme is both highly accurate and efficient while preserving mass and energy exactly. Numerical experiments are conducted to verify the accuracy and conservation properties of the method.<\/jats:p>","DOI":"10.3390\/axioms15030181","type":"journal-article","created":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:24:34Z","timestamp":1772447074000},"page":"181","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Structure-Preserving Scheme for the Space-Fractional Klein-Gordon-Schr\u00f6dinger System with the Invariant Energy Quadratization Method"],"prefix":"10.3390","volume":"15","author":[{"given":"Wenye","family":"Jiang","sequence":"first","affiliation":[{"name":"School of Science, Northeast Forestry University, Harbin 150040, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4993-7377","authenticated-orcid":false,"given":"Yu","family":"Li","sequence":"additional","affiliation":[{"name":"School of Science, Northeast Forestry University, Harbin 150040, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0674-5866","authenticated-orcid":false,"given":"Yan","family":"Fan","sequence":"additional","affiliation":[{"name":"Xingzhi College, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113993","DOI":"10.1016\/j.jcp.2025.113993","article-title":"Unconditionally stable explicit exponential methods for the Klein-Gordon-Schr\u00f6dinger equations","volume":"533","author":"Mei","year":"2025","journal-title":"J. Comput. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3131","DOI":"10.1103\/PhysRevE.62.3135","article-title":"Fractional quantum mechanics","volume":"62","author":"Laskin","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/S0375-9601(00)00201-2","article-title":"Fractional quantum mechanics and L\u00e9vy path integrals","volume":"268","author":"Laskin","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.matcom.2023.02.006","article-title":"Two efficient exponential energy-preserving schemes for the fractional Klein-Gordon-Schr\u00f6dinger equation","volume":"209","author":"Guo","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"056108","DOI":"10.1103\/PhysRevE.66.056108","article-title":"Fractional Schr\u00f6dinger equation","volume":"66","author":"Laskin","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2629","DOI":"10.1016\/j.camwa.2017.12.033","article-title":"Linearly implicit predictor-corrector methods for space-fractional reaction-diffusion equations with non-smooth initial data","volume":"75","author":"Khalil","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kilbas, A.A. (2006). Theory and Applications of Fractional Differential Equations, Elsevier.","DOI":"10.3182\/20060719-3-PT-4902.00008"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Meerschaert, M.M., and Sikorskii, A. (2019). Stochastic Models for Fractional Calculus, De Gruyter.","DOI":"10.1515\/9783110559149"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1110","DOI":"10.1016\/j.matcom.2021.07.003","article-title":"Energy-preserving scheme for the nonlinear fractional Klein-Gordon-Schr\u00f6dinger equation","volume":"190","author":"Wu","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1007\/s11075-019-00793-9","article-title":"Fast conservative numerical algorithms for the coupled fractional Klein-Gordon-Schr\u00f6dinger equation","volume":"84","author":"Li","year":"2020","journal-title":"Numer. Algorithms"},{"key":"ref_11","first-page":"348","article-title":"Conservative Fourier spectral method and numerical investigation of space fractional Klein-Gordon-Schr\u00f6dinger equation","volume":"350","author":"Wang","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1017\/S0308210511000746","article-title":"Positive solutions of the nonlinear Schr\u00f6dinger equation with the fractional Laplacian","volume":"142","author":"Felmer","year":"2012","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.cnsns.2016.04.020","article-title":"Fourier spectral method for higher order space fractional reaction-diffusion equations","volume":"40","author":"Pindza","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.cma.2016.10.041","article-title":"Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model","volume":"315","author":"Yang","year":"2017","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1629","DOI":"10.1007\/s10915-019-01001-5","article-title":"A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach","volume":"80","author":"Jiang","year":"2019","journal-title":"J. Sci. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"B135","DOI":"10.1137\/18M1213579","article-title":"Arbitrarily high-order unconditional energy stable schemes for thermodynamically consistent gradient flow models","volume":"42","author":"Gong","year":"2020","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.aml.2019.03.032","article-title":"A linear, symmetric and energy-conservative scheme for the space-fractional Klein-Gordon-Schr\u00f6dinger equations","volume":"95","author":"Wang","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"106091","DOI":"10.1016\/j.aml.2019.106091","article-title":"Novel high-order energy-preserving diagonally implicit Runge-Kutta schemes for nonlinear Hamiltonian ODEs","volume":"102","author":"Zhang","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"109055","DOI":"10.1016\/j.aml.2024.109055","article-title":"Mass and energy conservative high-order diagonally implicit Runge-Kutta schemes for nonlinear Schr\u00f6dinger equations","volume":"153","author":"Liu","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Feng, K., and Qin, M. (2010). Symplectic Geometric Algorithms for Hamiltonian Systems, Springer.","DOI":"10.1007\/978-3-642-01777-3"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1823","DOI":"10.1007\/s10543-022-00932-0","article-title":"Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems","volume":"62","author":"Lessig","year":"2022","journal-title":"BIT Numer. Math."},{"key":"ref_22","unstructured":"Hairer, E., N\u00f8rsett, S.P., and Wanner, G. (1993). Solving Ordinary Differential Equations I: Nonstiff Problems, Springer. [2nd ed.]."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1364\/OL.40.001117","article-title":"Fractional Schr\u00f6dinger equation in optics","volume":"40","author":"Longhi","year":"2015","journal-title":"Opt. Lett."},{"key":"ref_24","unstructured":"Boyd, R.W. (2008). Nonlinear Optics, Academic Press. [3rd ed.]."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kivshar, Y.S., and Agrawal, G.P. (2003). Optical Solitons: From Fibers to Photonic Crystals, Academic Press.","DOI":"10.1016\/B978-012410590-4\/50012-7"},{"key":"ref_26","unstructured":"Allen, L., and Eberly, J.H. (1975). Optical Resonance and Two-Level Atoms, Wiley."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/3\/181\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T05:33:22Z","timestamp":1772775202000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/3\/181"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,1]]},"references-count":26,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["axioms15030181"],"URL":"https:\/\/doi.org\/10.3390\/axioms15030181","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,1]]}}}