{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:48:42Z","timestamp":1765212522647,"version":"3.46.0"},"reference-count":42,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T00:00:00Z","timestamp":1764374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundations of Shandong","award":["No. ZR2024MA017","ZR2023MA062","ZR202204010001"],"award-info":[{"award-number":["No. ZR2024MA017","ZR2023MA062","ZR202204010001"]}]},{"name":"Science and Technology Plan Project of Dezhou","award":["No. 2021dzkj1638"],"award-info":[{"award-number":["No. 2021dzkj1638"]}]},{"name":"Research Platform Project of Dezhou University","award":["No. 2023XKZX024"],"award-info":[{"award-number":["No. 2023XKZX024"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we propose two methods for parameter estimation in stochastic Korteweg\u2013de Vries (KdV) equations with unknown parameters. Both methods are based on the numerical discretization of the stochastic KdV equation. Moreover, we further propose an extrapolation-based approach to improve the accuracy of parameter estimation. In addition, for the deterministic case, the convergence and conservation of the fully discrete schemes are analyzed. Both our theoretical analysis and numerical tests indicate the efficiency of the proposed methods for the KdV equations considered.<\/jats:p>","DOI":"10.3390\/axioms14120884","type":"journal-article","created":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:41:41Z","timestamp":1765212101000},"page":"884","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Parameter Estimation for Stochastic Korteweg\u2013de Vries Equations"],"prefix":"10.3390","volume":"14","author":[{"given":"Zhenyu","family":"Lang","sequence":"first","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"},{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4495-4098","authenticated-orcid":false,"given":"Xiuling","family":"Yin","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanqin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yaru","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"},{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1080\/14786449508620739","article-title":"On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary wave","volume":"39","author":"Korteweg","year":"1895","journal-title":"Philos. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"115914","DOI":"10.1016\/j.cam.2024.115914","article-title":"Linearly-fitted energy-mass-preserving schemes for Korteweg\u2013de Vries equations","volume":"448","author":"Liu","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"ref_3","first-page":"126101","article-title":"Solving the Korteweg-de Vries equation with Hermite-based finite differences","volume":"401","author":"Abrahamsen","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1007\/s10092-021-00443-4","article-title":"On a high order Gaussian radial basis function generated Hermite fnite diference method and its application","volume":"58","author":"Soleymani","year":"2021","journal-title":"Calcolo"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.apnum.2019.07.001","article-title":"A numerical method to compute the scattering solution for the KdV equation","volume":"149","author":"Fermo","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Williams, K., and Akers, B. (2023). Numerical Simulation of the Korteweg\u2013de Vries Equation with Machine Learning. Mathematics, 11.","DOI":"10.3390\/math11132791"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"106256","DOI":"10.1016\/j.aml.2020.106256","article-title":"Soliton and breather solutions for a fifth-order modified KdV equation with a nonzero background","volume":"104","author":"Liu","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/S0167-2789(99)00072-X","article-title":"Numerical simulation of the stochastic Korteweg\u2013de Vries equation","volume":"134","author":"Debussche","year":"1999","journal-title":"Phys. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.physleta.2004.05.026","article-title":"Exact solutions for Wick-type stochastic coupled KdV equations","volume":"327","author":"Xie","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"129004","DOI":"10.1016\/j.physa.2023.129004","article-title":"The stochastic Korteweg\u2013de Vries equation with loss and non-uniformity terms","volume":"625","author":"Zhao","year":"2023","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1023\/A:1013549126956","article-title":"Long-time dynamics of Korteweg\u2013de Vries solitons driven by random perturbations","volume":"105","author":"Garnier","year":"2001","journal-title":"J. Stat. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1121\/1.2395914","article-title":"Internal solitons in the ocean and their effect on underwater sound","volume":"121","author":"Apel","year":"2007","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1088\/0305-4470\/23\/7\/014","article-title":"The stochastic, damped KdV equation","volume":"23","author":"Herman","year":"1990","journal-title":"J. Phys. A Gen. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00033-019-1165-4","article-title":"Averaging principle for Korteweg\u2013de Vries equation with a random fast oscillation","volume":"70","author":"Gao","year":"2019","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"123560","DOI":"10.1016\/j.physa.2019.123560","article-title":"Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg\u2013de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma","volume":"544","author":"Seadawy","year":"2020","journal-title":"Phys. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1757","DOI":"10.1063\/1.4756515","article-title":"Stochastic multisymplectic integrator for stochastic KdV equation","volume":"1479","author":"Jiang","year":"2012","journal-title":"AIP Conf. Proc."},{"key":"ref_17","first-page":"5552","article-title":"Multi-symplectic methods for the Ito-type coupled KdV equation","volume":"218","author":"Chen","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apnum.2024.06.001","article-title":"A structure-preserving local discontinuous Galerkin method for the stochastic KdV equation","volume":"204","author":"Liu","year":"2024","journal-title":"Appl. Numer. Math."},{"key":"ref_19","first-page":"60","article-title":"Nonlinearization of spectral problems for the perturbation KdV systems","volume":"296","author":"Ma","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Lang, Z., Yin, X., Liu, Y., Chen, Z., and Kong, S. (2024). Combined compact symplectic schemes for solving good Boussinesq equations. Axioms, 13.","DOI":"10.20944\/preprints202406.1509.v1"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1007\/s10915-023-02219-0","article-title":"Structure-Preserving Combined High-Order Compact Schemes for Multiple Order Spatial Derivatives Differential Equations","volume":"96","author":"Wang","year":"2023","journal-title":"J. Sci. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/j.apnum.2018.09.005","article-title":"A novel kind of efficient symplectic scheme for Klein-Gordon-Schr\u00f6dinger equation","volume":"135","author":"Kong","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_23","first-page":"193","article-title":"Multi-symplectic Fourier Pseudospectral Method for the Nonlinear schro\u00a8 dinger equation","volume":"12","author":"Jing","year":"2001","journal-title":"Elect. Transa. Numer. Anal."},{"key":"ref_24","first-page":"31","article-title":"Multi-symplectic Fourier Pseudospectral Method for the Nonlinear schro\u00a8 dinger equations with wave operator","volume":"25","author":"Wang","year":"2007","journal-title":"J. Comput. Math."},{"key":"ref_25","first-page":"275","article-title":"Symplectic Fourier Pseudo-spectral Schemes for Klein-Gordon- Schrodinger equation","volume":"28","author":"Wang","year":"2011","journal-title":"China J. Comput. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"107682","DOI":"10.1016\/j.aml.2021.107682","article-title":"Energy-preserving splitting methods for charged-particle dynamics in a normal or strong magnetic field","volume":"124","author":"Li","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.apnum.2018.09.011","article-title":"Stochastic multi-symplectic Runge-Kutta methods for stochastic Hamiltonian PDEs","volume":"135","author":"Zhang","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1137\/19M1306919","article-title":"Asymptotically-Preserving Large Deviations Principles by Stochastic Symplectic Methods for a Linear Stochastic Oscillator","volume":"59","author":"Chen","year":"2021","journal-title":"Siam. J. Numer. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Evans, L. (2013). An Introduction to Stochastic Differential Equations, American Mathematical Society.","DOI":"10.1090\/mbk\/082"},{"key":"ref_30","first-page":"529","article-title":"Some methods of parameter estimation for stochastic differential equations","volume":"34","author":"Cai","year":"2017","journal-title":"J. Univ. Chin. Acad. Sci."},{"key":"ref_31","first-page":"136","article-title":"Parameter estimation for some type of stochastic partial differential equation in the infinite dimensional space","volume":"42","author":"Zhang","year":"2015","journal-title":"J. Zhejiang Univ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1016\/S0378-3758(00)00196-8","article-title":"Bayes estimation for some stochastic partial differential equations","volume":"91","author":"Rao","year":"2000","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1177\/0008068320030301","article-title":"Estimation for Some Stochastic Partial Differential Equations Based on Discrete Observations II","volume":"54","author":"Rao","year":"2003","journal-title":"Calcutta Stat. Assoc. Bull."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4033","DOI":"10.1016\/j.jspi.2008.02.016","article-title":"On asymptotic properties of the parameter estimator for a type of SPDE","volume":"138","author":"Zhang","year":"2008","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1142\/S0219493710003091","article-title":"Parameter estimation for SPDEs with multiplicative fractional noise","volume":"10","author":"Cialenco","year":"2010","journal-title":"Stoch. Dyn."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.apnum.2017.04.006","article-title":"A second order operator splitting numerical scheme for the \u201cgood\u201d Boussinesq equation","volume":"119","author":"Zhang","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_37","unstructured":"Boyd, J. (2001). Chebyshev and Fourier Spectral Methods, Dover. [2nd ed.]."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussani, M.Y., Quarteroni, A., and Zang, T.A. (2007). Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer.","DOI":"10.1007\/978-3-540-30728-0"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Hesthaven, J., Gottlieb, S., and Gottlieb, D. (2007). Spectral Methods for Time-Dependent Problems, Cambridge University Press.","DOI":"10.1017\/CBO9780511618352"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/0723001","article-title":"The exponential accuracy of Fourier and Chebyshev differencing methods","volume":"23","author":"Tadmor","year":"1986","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_41","unstructured":"Bishwal, J. (1923). Parameter Estimation in Stochastic Differential Equations, Springer. Lecture Notes in Mathematics."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1103\/PhysRevLett.19.1095","article-title":"Method for Solving Korteweg-Devries Equation","volume":"19","author":"Gardner","year":"1967","journal-title":"Phys. Rev. Lett."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/884\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:43:51Z","timestamp":1765212231000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/884"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,29]]},"references-count":42,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["axioms14120884"],"URL":"https:\/\/doi.org\/10.3390\/axioms14120884","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,29]]}}}