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Nonlinear Analysis and Computer Simulations: New Horizons in Theory and Applications

  • Carlo Cattani 1,2,3,4

Received: 26 Nov 2025 | Accepted: 05 Dec 2025 | Published: 05 Jan 2026

References 

  • 1.

    Adomian, G. Nonlinear Stochastic Operator Equations; Kluwer Academic Publishers: Alphen aan den Rijn, The Netherlands, 1986.

  • 2.

    Adomian, G. Solving Frontier Problems of Physics: The Decomposition Method; Kluwer Academic Publishers: Alphen aan den Rijn, The Netherlands, 1994.

  • 3.

    Jiong, S. Auxiliary equation method for solving nonlinear partial differential equations. Phys. Lett. A 2003, 309, 387–396.

  • 4.

    Fendzi-Donfack, E.; Temgoua, G.W.K.; Djoufack, Z.I.; et al. Exotical solitons for an intrinsic fractional circuit using the sine–cosine method. Chaos Solitons Fractals 2022, 160, 112253.

  • 5.

    Zhu, D.S. Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method. Chaos Solitons Fractals 2007, 34, 1608–1612.

  • 6.

    Zhang, S.; Xia, T.-C. A generalized auxiliary equation method and its application to (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations. J. Phys. A Math. Theor. 2007, 40, 227–248.

  • 7.

    Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA, 1982.

  • 8.

    He, J.-H. Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 2005, 6, 207–208.

  • 9.

    He, J.-H. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 2005, 26, 695–700.

  • 10.

    Liao, S.J. The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China, 1992.

  • 11.

    Liao, S. Notes on the homotopy analysis method: Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simulat. 2009, 14, 983–997.

  • 12.

    Liao, S. Homotopy Analysis Method in Non-Linear Differential Equations; Springer: Berlin, Germany, 2012.

  • 13.

    He, J.-H. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals 2004, 19, 847–851.

  • 14.

    He, J.-H. Variational approach to (2+1)-dimensional dispersive long water equations. Phys. Lett. A 2005, 335, 182–184.

  • 15.

    Liu, H.-M. Variational approach to nonlinear electrochemical system. Int. J. Nonlinear Sci. Numer. Simul. 2004, 5, 95–96.

  • 16.

    Russell, J.S. Report on Waves. In Proceedings of the 14th Meeting of the British Association for the Advancement of Science, New York, NY, USA, 23–28 September 1844; pp. 311–390.

  • 17.

    Korteweg, G.; De Vries, D.J.. XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Lond. Edinb. Dublin Philos. Mag. J. Sci. 1895, 39, 422–443.

  • 18.

    Adlam, J.H.; Allen, J.E. The structure of strong collision-free hydromagnetic waves. Philos. Mag. 1958, 3, 448–455.

  • 19.

    Adlam, J.H.; Allen, J.E. Collision-free Hydromagnetic Disturbances of Large Amplitude in a Plasma. Proc. Phys. Soc. 1960, 75, 640–648.

  • 20.

    Zabusky, N.J.; Kruskal, M.D. Interaction of ‘Solutions’ in a Collisionless Plasms and the Recurrence of Intial States. Phys. Rev. Lett. 1965, 15, 240–243.

  • 21.

    Ablowitz, M.J.; Clarkson, P.A. Solitons, Nonlinear Evolution Equations and Inverse Scattering; Cambridge University Press: Cambridge, UK, 1991.

  • 22.

    Wadati, M. Wave propagation in nonlinear lattice. J. Phys. Soc. Jpn. 1975, 38, 673–680.

  • 23.

    Matveev, V.B.; Salle, M.A. Darboux Transformation and Solitons; Springer Series in Nonlinear Dynamics; Springer: Berlin, Germany, 1991.

  • 24.

    Zait, R.A. Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations. Chaos Solitons Fractals 2003, 15, 673–678.

  • 25.

    Hirota, R. Exact solution of the korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 1971, 27, 1192–1194.

  • 26.

    Hirota, R. The Direct Method in Soliton Theory; Cambridge University Press: Cambridge, UK, 2004.

  • 27.

    Weiss, J.; Tabor, M.; Carnevale, G. The Painlevé property for partial differential equations. J. Math. Phys. 1982, 24, 522–526.

  • 28.

    Wang, M.L.; Li, X.Z.; Zhang, J. The (G′G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008, 372, 417–423.

  • 29.

    Zhang, H. New application of the (G′G)-expansion method. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 3220–3225.

  • 30.

    Wen-An, L.; Hao, C.; Guo-Cai, Z. The (ω/g)-expansion method and its application to Vakhnenko equation. Chin. Phys. B 2009, 18, 400–404.

  • 31.

    Zuo, J.-M.; Zhang, Y.-M. Application of the G′G-expansion method to solve coupled MKdV equations and coupled Hirota–Satsuma coupled KdV equations. Appl. Math. Comput. 2011, 217, 5936–5941.

  • 32.

    Zhang, J.; Jiang, F.; Zhao, X. An improved (G’/G)-expansion method for solving nonlinear evolution equations. Int. J. Comput. Math. 2010, 87, 1716–1725.

  • 33.

    Kudryashov, N.A. One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 2248–2253.

  • 34.

    Roshid, H.O.; Azizur Rahman, M. The exp(-Φ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results Phys. 2014, 4, 150–155.

  • 35.

    Khan, K.; Akbar, M.A. The exp(−Φ(ξ))-expansion method for finding travelling wave solutions of Vakhnenko–Parkes equation. Int. J. Dyn. Syst. Differ. 2014, 5, 72–83.

  • 36.

    Chen, H.-T.; Zhang, H.-Q. New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation. Chaos Solitons Fractals 2004, 20, 765–769.

  • 37.

    Zhang, D. Doubly periodic solutions of the modified Kawahara equation. Chaos Solitons Fractals 2005, 25, 1155–1160.

  • 38.

    Li, W.-W.; Tian, Y.; Zhang, Z. F-expansion method and its application for finding new exact solutions to the sine–Gordon and sinh-Gordon equations. Appl. Math. Comput. 2012, 219, 1135–1143.

  • 39.

    He, J.-H.; Wu, X.-H. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals, 2006, 30, 700–708.

  • 40.

    Xu, F. Application of Exp-function method to Symmetric Regularized Long Wave (SRLW) equation. Phys. Lett. A 2008, 372, 252–257.

  • 41.

    Ma, W.; Huang, T.; Zhang, Y. A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 2010, 82, 065003.

  • 42.

    Fan, E.G. Extended tanh-function method and its application to nonlinear equations. Phys. Lett. A 2000, 277, 212–218.

  • 43.

    Liu, S.; Fu, Z.; Liu, S.; et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289, 69–74.

  • 44.

    Fu, Z.; Liu, S.; Liu, S.; et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A 2001, 290, 72–76.

  • 45.

    Liu, G.-T.; Fan, T.-Y. New applications of developed Jacobi elliptic function expansion methods. Phys. Lett. A 2005, 345, 161–166.

  • 46.

    Zhang, H. Extended Jacobi elliptic function expansion method and its applications. Commun. Nonlinear Sci. Numer. Simul. 2007, 12, 627–635.

  • 47.

    Wazwaz, A.M. Compactons in a Class of Nonlinear Dispersive Equations. Math. Comput. Model. 2003, 37, 333–341.

  • 48.

    Wazwaz, A.M. The sine-cosine method for obtaining solutions with compact and noncompact structures. Appl. Math. Comput. 2004, 159, 559–576.

  • 49.

    Baskonus, H.M. New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 2016, 86, 177–183.

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Cattani, C. Nonlinear Analysis and Computer Simulations: New Horizons in Theory and Applications. Nonlinear Analysis and Computer Simulations 2026, 1 (1), 1.
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