THERMAL SCIENCE

International Scientific Journal

Meng-Zhu Liu

,

Xiao-Qian Zhu

,

Xiao-Qun Cao

,

Bai-Nian Liu

,

Ke-Cheng Peng

INTERNAL SOLITARY WAVES IN THE OCEAN BY SEMI-INVERSE VARIATIONAL PRINCIPLE

ABSTRACT
Internal solitary waves are very common physical phenomena in the ocean, which play an important role in the transport of marine matter, momentum and energy. The non-linear Schrodinger equation is suitable for describing the deep-sea internal wave propagation. Firstly, by designing skillfully, the trial-Lagrange functional, variational principles are successfully established for the non-linear Schrodinger equation by the semi-inverse method. Secondly, the constructed var-iational principle is proved by minimizing the functionals with the calculus of variations. Finally, different kinds of internal solitary waves are obtained by the semi-inverse variational principle for the non-linear Schrodinger equation.
KEYWORDS
internal solitary wave, variational principle, semi-inverse method, non-linear Schrodinger equation
PAPER SUBMITTED: 2020-10-12
PAPER REVISED: 2021-10-01
PAPER ACCEPTED: 2021-10-01
PUBLISHED ONLINE: 2022-07-16
DOI REFERENCE: https://doi.org/10.2298/TSCI2203517L
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 3, PAGES [2517 - 2525]
REFERENCES
  1. Jiang, Z. H., et al., Ocean Internal Waves Interpreted as Oscillation Travelling Waves in Consideration of Ocean Dissipation, Chin. Phys. B, 23 (2014), 5, 050302
  2. Lee, C. Y, Beardsley R. C., The Generation of Long Nonlinear Internal Waves in a Weakly Stratified Shear flow, Journal of Geophysical Research, 79 (1974), 3, pp. 453-462
  3. Wang, Z., Zhu Y. K., Theory, Modelling and Computation of Nonlinear Ocean Internal Waves, Chinese Journal of Theoretical and Applied Mechanics, 51 (2019), 6, pp. 1589-1604
  4. Karunakar, P., Chakraverty, S., Effect of Coriolis Constant on Geophysical Korteweg-de Vries Equation, Journal of Ocean Engineering and Science, 4 (2019), 2, pp. 113-121
  5. Kaya, D., Explicit and Numerical Solutions of Some Fifth-order KdV Equation by Decomposition Method, Appl Math Comput, 144 (2003), 2-3, pp. 353-363
  6. Kaya, D., EI-Sayed, S. M., On a Generalized Fifth-Order KdV Equations, Phys. Lett. A, 310 (2003), 1, pp. 44-51
  7. Zhang, Y., Chen, D. Y., The Novel Multi Solitary Wave Solution to the Fifth-Order KdV Equation, Chin. Phys. B, 10 (2004), 10, pp. 1606-1610
  8. Wazwaz, A. M., A Study on Compacton-Like Solutions for the Modified KdV and Fifth Order KdV-Like Equations, Appl. Math. Comput., 147 (2004), 2, pp. 439-447
  9. He, J.-H. Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, J. Appl. Comput. Mech., 6 (2020), 4, pp. 735-740
  10. Li, J., et al., Simulation Investigation on the Internal Wave via the Analytical Solution of Korteweg-de Vries Equation (in Chinese), Marine Science Bulletin, 30 (2011), 1, pp. 23-28
  11. Benjamin, B. T., Internal Waves of Permanent form in Fluids of Great Depth, Journal of Fluid Mechan-ics, 29 (1967), 3, pp. 559-592
  12. Ono, H. Algebraic Solitary Waves in Stratified Fluids, Journal of the Physical Society of Japan, 39 (1975), 4, pp. 1082-1091
  13. Kubota, T., et al., Propagation of Weakly Non-linear Internal Waves in a Stratified Fluid of Finite Depth, AIAA Journal of Hydronautics, 12 (1978), 4, pp. 157-165
  14. Choi, W., Camassa, R., Fully Non-Linear Internal Waves in a Two-Fluid System, Journal of Fluid Me-chanics, 396 (1999), Oct., pp. 1-36
  15. Song, S. Y., et al., Non-Linear Schrödinger Equation for Internal Waves in Deep Sea, Acta Physica Sini-ca, 59 (2010), 2, pp. 1123-1129
  16. Wazwaz, A. M., Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing, 2009
  17. Wazwaz, A. M., Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press, Beijing, 2011
  18. He, J. H., Exp-Function Method for Fractional Differential Equations, Int. J. Nonlinear Sci. Numer. Sim-ul., 14 (2013), 6, pp. 363-366
  19. He, J.‐H., et al., Homotopy Perturbation Method for the Fractal Toda Oscillator, Fractal Fract., 5 (2021), 3, 93
  20. Wu, Y., Variational Approach to Higher-Order Water-Wave Equations, Chaos Solitons Fractals, 32 (2007), 1, pp.195-203.
  21. Gazzola, F., et al., Variational Formulation of the Melan Equation, Math. Methods Appl. Sci., 41 (2018), 3, pp. 943-951
  22. Durgun, D. D., Fractional Variational Iteration Method for Time-Fractional Nonlinear Functional Partial Differential Equation Having Proportional Delays, Thermal Science, 22 (2018), Suppl. 1, pp. S33-S46
  23. He, C. H., et al. Hybrid Rayleigh-van der Pol-Duffing Oscillator: Stability Analysis and Controller, Journal of Low Frequency Noise Vibration and Active Control, 41 (2022), 1, pp. 244-268
  24. Tian, D., et al., Fractal N/MEMS: from Pull-In Instability to Pull-In Stability, Fractals, 29 (2021), 2, 2150030
  25. Tian, D., He, C. H., A Fractal Micro-Electromechanical System and Its Pull-In Stability, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, pp. 1380-1386
  26. Li, X. X., He, C. H., Homotopy Perturbation Method Coupled with the Enhanced Perturbation Method, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1399-1403
  27. He, J. H. Variational Iteration Method - Some Recent Results and New Interpretations, Journal of Com-putational and Applied Mathematics, 207 (2007), 1, pp. 3-7
  28. He, J. H., Wu, X. H., Variational Iteration Method: New Development and Applications, Computers & Mathematics with Applications, 54 (2007), 7-8, pp. 894-881
  29. Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Itera-tion Method, Thermal Science, 17 (2013), 2, pp. 625-628
  30. He, J. H.; et al., Dynamic Pull-In for Micro-Electromechanical Device with a Current-Carrying Conduc-tor, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 2, pp. 1059-1066
  31. Malomed, B. A., Variational Methods in Nonlinear fiber Optics and Related fields, Prog. Opt., 43 (2002), 71, pp. 71-193
  32. Chong, C., Pelinovsky, D. E., Variational Approximations of Bifurcations of Asymmetric Solitons in Cubic-Quintic Nonlinear Schrödinger Lattices, Discret. Contin. Dyn. Syst., 4 (2011), 5, pp. 1019-1031
  33. Kaup, D. J., Variational Solutions for the Discrete Nonlinear Schrödinger Equation, Math. Comput. Sim-ul., 69 (2005), 3-4, pp. 322-333
  34. Putri, N. Z., et al., Variational Approximations for Intersite Soliton in a Cubic-Quintic Discrete Nonline-ar Schrödinger Equation, J. Phys. Conf. Ser., 1317 (2019), 1, 012015
  35. He, J. H., Variational Principles for Some Nonlinear Partial Differential Equations with Variable Coeffi-cients, Chaos Solitons Fractals, 19 (2004), 4, pp. 847-851
  36. He, J. H. Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics with Emphasis on Turbomachinery Aerodynamics, International Journal of Turbo & Jet-Engines, 14 (1997), 1, pp. 23-28
  37. He, J. H., et al. On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathematics Letters, 119 (2021), Sept., 107199
  38. Wang, K. J., Generalized Variational Principle and Periodic Wave Solution to the Modified Equal width-Burgers Equation in Nonlinear Dispersion Media, Physics Letters A, 419 (2021), Dec., 127723
  39. He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199
  40. Cao, X. Q., et al., Variational Theory for (2+1)-Dimensional Fractional Dispersive Long Wave Equa-tions, Thermal Science, 25 (2021), 2, pp. 1277-1285
  41. Wang, K. J., Zhang, P. L., Investigation of the Periodic Solution of the Time-Space Fractional Sasa-Satsuma Equation Arising in the Monomode Optical Fibers, EPL, 137 (2021), 6, 62001
  42. Khan, Y., Fractal Higher-Order Dispersions Model and Its Fractal Variational Principle Arising in the Field of Physcial Process, Fluctuation and Noise Letters, 20 (2021), 4, 2150034
  43. He, J. H., Some Asymptotic Methods for Strongly Nonlinear wave Equation, International Journal of Modern Physics B, 20 (2006), 10, pp. 1141-1199
  44. Li, Y., He, C. H., A Short Remark on Kalaawy's Variational Principle for Plasma, International Journal of Numerical Methods for Heat & Fluid Flow, 27 (2017), 10, pp. 2203-2206
  45. He, J. H., et al., A Fractal Modification of Chen-Lee-Liu Equation and Its Fractal Variational Principle, International Journal of Modern Physics B, 35 (2021), 21, 2150214
  46. He, J. H., Maximal Thermo-geometric Parameter in a Nonlinear Heat Conduction Equation, Bulletin of the Malaysian Mathematical Sciences Society, 39 (2016), 2, pp. 605-608
  47. Cao, X. Q., Variational Principles for Two Kinds of Extended Korteweg-de Vries Equations, Chin. Phys. B, 20 (2011), 9, pp. 94-102
  48. Cao, X. Q., Generalized Variational Principles for Boussinesq Equation Systems, Acta Phys. Sin., 60 (2011), 8, pp. 105-113
  49. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950132
  50. He, J. H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), May, 104104
  51. Wang, K. L., et al., Physical Insight of Local Fractional Calculus and Its Application to Fractional KdV-Burgers-Kuramoto Equation, Fractals, 27 (2019), 7, 1950122
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Internal solitary waves in the ocean by semi-inverse variational principle

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