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All issues Volume 17 (2022) Math. Model. Nat. Phenom., 17 (2022) 43 References
Open Access
Issue
Math. Model. Nat. Phenom.
Volume 17, 2022
Article Number 43
Number of page(s) 13
DOI https://doi.org/10.1051/mmnp/2022044
Published online 09 November 2022
  1. M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons. Cambridge University Press, Cambridge (2011). [Google Scholar]
  2. M.J. Ablowitz, G. Biondini and S. Blair, Nonlinear Schrödinger equations with mean terms in nonresonant multidimensional quadratic materials. Phys. Rev. E 63 (2001) 046605. [CrossRef] [Google Scholar]
  3. Sh. Amiranashvili and A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses. Phys. Rev. A 82 (2010) 013812. [CrossRef] [Google Scholar]
  4. Sh. Amiranashvili, R. Čiegis and M. Radziunas, Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinet. Relat. Mod. 8 (2015) 215–234. [CrossRef] [Google Scholar]
  5. D.J. Benney and A.C. Newell, The propagation of nonlinear wave envelopes. J. Math. Phys. 46 (1967) 133–139. [CrossRef] [MathSciNet] [Google Scholar]
  6. T.J. Bridges, M.D. Groves and D.P. Nicholls (eds.), Lectures on the Theory of Water Waves. Cambridge University Press, Cambridge (2016). [Google Scholar]
  7. T.J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A 284 (2001) 184–193. [CrossRef] [MathSciNet] [Google Scholar]
  8. A.V. Buryak, P. Di Trapani, D.V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370 (2002) 63–235. [CrossRef] [MathSciNet] [Google Scholar]
  9. D. Cline, Variation Principles in Classical Mechanics, University of Rochester, Rochester, 2nd edn. (2018). [Google Scholar]
  10. W. Craig, P. Guyenne and C. Sulem, A Hamiltonian approach to nonlinear modulation of surface water waves. Wave Motion 47 (2010) 552–563. [CrossRef] [MathSciNet] [Google Scholar]
  11. W. Craig, P. Guyenne and C. Sulem, Hamiltonian higher-order nonlinear Schrödinger equations for broader-banded waves on deep water. Eur. J. Mech. B/Fluids 32 (2012) 22–31. [CrossRef] [MathSciNet] [Google Scholar]
  12. W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5 (1992) 497–522. [Google Scholar]
  13. J. Cuevas-Maraver, P.G. Kevrekidis and F. Williams, The Sine–Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics. Springer, New York (2014). [Google Scholar]
  14. T. Dauxois, M. Peyrard and C.R. Willis, Localized breather-like solution in a discrete Klein–Gordon model and application to DNA. Physica D 57 (1992) 267–282. [CrossRef] [MathSciNet] [Google Scholar]
  15. D. Dutykh, D. Clamond and M. Chhay, Serre-type equations in deep water. Math. Model. Nat. Phenom. 12 (2017) 23–40. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  16. I.S. Gandzha and Yu.V. Sedletsky, A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation. Nonlin. Dyn. 98 (2019) 359–374. [CrossRef] [Google Scholar]
  17. P. Germain and F. Pusateri, Quadratic Klein–Gordon equations with a potential in one dimension. Forum Math. Pi 10 (2022) e17. [CrossRef] [Google Scholar]
  18. H. Goldstein, C. Poole and J. Safko, Classical Mechanics. Addison Wesley, San Francisco, 3rd edn. (2001). [Google Scholar]
  19. O. Gramstad and K. Trulsen, Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth. J. Fluid Mech. 670 (2011) 404–426. [CrossRef] [MathSciNet] [Google Scholar]
  20. P. Guyenne, D.P. Nicholls and C. Sulem (eds.), Hamiltonian Partial Differential Equations and Applications. Springer, New York (2015). [Google Scholar]
  21. E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos. Cambridge University Press, Cambridge (1990). [Google Scholar]
  22. C.K.R.T. Jones, R. Marangell, P.D. Miller and R.G. Plaza, Spectral and modulational stability of periodic wavetrains for the nonlinear Klein–Gordon equation. J. Differ. Equ. 257 (2014) 4632–4703. [CrossRef] [Google Scholar]
  23. A.G. Kalocsai and J.W. Haus, Nonlinear Schrödinger equation for opical media with quadratic nonlinearity. Phys. Rev. A 49 (1994) 574–585. [CrossRef] [PubMed] [Google Scholar]
  24. P.G. Kevrekidis and J. Cuevas-Maraver (eds.), A Dynamical Perspective on the ϕ4 Model: Past, Present, and Future. Springer, Cham (2019). [Google Scholar]
  25. C. Lämmerzahl, The pseudodifferential operator square root of the Klein–Gordon equation. J. Math. Phys. 34 (1993) 3918–3932. [CrossRef] [MathSciNet] [Google Scholar]
  26. H. Leblond and D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523 (2013) 61–126. [CrossRef] [MathSciNet] [Google Scholar]
  27. V.P. Lukomsky and I.S. Gandzha, Two-parameter method for describing the nonlinear evolution of narrow-band wave trains. Ukr. J. Phys. 54 (2009) 207–215. [Google Scholar]
  28. A.C. Newell, Solitons in Mathematics and Physics. Society for Industrial and Applied Mathematics, Philadelphia (1985). [Google Scholar]
  29. M. Onorato, S. Residori and F. Baronio (eds.), Rogue and Shock Waves in Nonlinear Dispersive Media. Springer, Cham (2016). [Google Scholar]
  30. R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland, Amsterdam (1987). [Google Scholar]
  31. R. Sassaman and A. Biswas, Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations. Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 3239–3246. [CrossRef] [Google Scholar]
  32. A. Scott, A nonlinear Klein–Gordon equation. Am. J. Phys. 37 (1969) 52–61. [CrossRef] [Google Scholar]
  33. Yu.V. Sedletsky, The fourth-order nonlinear Schrodinger equation for the envelope of Stokes waves on the surface of a finite- depth fluid. JETP 97 (2003) 180–193. [CrossRef] [Google Scholar]
  34. Yu.V. Sedletsky, A fifth-order nonlinear Schrödinger equation for waves on the surface of finite-depth fluid. Ukr. J. Phys. 66 (2021) 41–54. [CrossRef] [Google Scholar]
  35. Yu.V. Sedletsky and I.S. Gandzha, A sixth-order nonlinear Schrödinger equation as a reduction of the nonlinear Klein–Gordon equation for slowly modulated wave trains. Nonlin. Dyn. 94 (2018) 1921–1932. [CrossRef] [Google Scholar]
  36. Yu.V. Sedletsky and I.S. Gandzha, Relationship between the Hamiltonian and non-Hamiltonian forms of a fourth-order nonlinear Schrödinger equation. Phys. Rev. E 102 (2020) 202202. [Google Scholar]
  37. I.T. Selezov, Yu.G. Kryvonos and I.S. Gandzha, Spectral methods in the theory of wave propagation, in Wave Propagation and Diffraction: Mathematical Methods and Applications. Foundations in Engineering Mechanics series, Springer, Singapore (2018) 25–75. [CrossRef] [Google Scholar]
  38. A.S. Sharma and B. Buti, Envelope solitons and holes for sine–Gordon and nonlinear Klein–Gordon equations. J. Phys. A: Math. Gen. 9 (1976) 1823–1826. [CrossRef] [Google Scholar]
  39. Sirendaoreji, Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations. Phys. Lett. A 363 (2007) 440–447. [CrossRef] [MathSciNet] [Google Scholar]
  40. G. Sterman, An Intorduction to Quantum Field Theory. Cambridge University Press, Cambridge (1993). [Google Scholar]
  41. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999). [Google Scholar]
  42. E. Tobisch (ed.), New Approaches to Nonlinear Waves. Springer, Cham (2016). [Google Scholar]
  43. A.-M. Wazwaz, Compactons, solitons and periodic solutions for some forms of nonlinear Klein–Gordon equations. Chaos Solit. Fract. 28 (2006) 1005–1013. [CrossRef] [Google Scholar]
  44. V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (1968) 190–194. [Google Scholar]
  45. V.E. Zakharov, V.S. L’vov and G. Falkovich, Kolmogorov Spectra of Turbulence I. Wave Turbulence. Springer, Berlin (1992). [Google Scholar]

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