Variational Principles and Solitary Wave Solutions of Generalized ‎Nonlinear Schrödinger Equation in the Ocean

Document Type : Research Paper


1 College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China

2 College of Computer, National University of Defense Technology, Changsha 410073, China‎


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. Because the generalized nonlinear Schrödinger equation can well explain the effects of nonlinearity and dispersion in the ocean, it is more suitable for describing the deep-sea internal wave propagation and evolution than other mathematical models. At first, by designing skillfully the trial-Lagrange functional, different kinds of variational principles are successfully established for a generalized nonlinear Schrödinger equation by the semi-inverse method. Then, the constructed variational principles are proved correct by minimizing the functionals with the calculus of variations. Furthermore, some kinds of internal solitary wave solutions are obtained and demonstrated by semi-inverse variational principle for the generalized nonlinear Schrödinger equation.


Main Subjects

[1] Jiang, Z.H., Huang, S.X., You, X. B., Ocean internal waves interpreted as oscillation travelling waves in consideration of ocean dissipation, Chinese Physics B, 23, 2014, 52-59.
[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, 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, 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, 113-121.
[5] Kaya, D., Explicit and Numerical Solutions of Some Fifth-order KdV Equation by Decomposition Method, Applied Mathematics and Computation, 144, 2003, 353-363.
[6] Wazwaz, A. M., A Study on Compacton-Like Solutions for the Modified KdV and Fifth Order KdV-Like Equations, Applied Mathematics and Computation, 147, 2004, 439-447.
[7] Wazwaz, A.M., Helal, M.A., Variants of the Generalized Fifth-Order KdV Equation with Compact and Noncompact Structures, Chaos Solitons and Fractals, 21, 2004, 579-589.
[8] Li, J., Gu X.F., Yu T., Sun Y. Simulation investigation on the internal wave via the analytical solution of Korteweg-de Vries equation, Marine Science Bulletin, 30, 2011, 23-28.
[9] Benjamin, B.T., Internal waves of permanent form in fluids of great depth, Journal of Fluid Mechanics, 29(03), 1967, 559-592.
[10] Hiroaki, O., Algebraic Solitary Waves in Stratified Fluids, Journal of the Physical Society of Japan, 39, 1975, 1082-1091.
[11] Kubota, T., Ko, D., Dobbs, L., Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth, AIAA Journal of Hydronautics, 12, 1978, 157-165.
[12] Choi, W., Camassa, R., Fully nonlinear internal waves in a two-fluid system, Journal of Fluid Mechanics, 396, 1999, 1-36.
[13] Song, S.Y., Wang, J., Meng, J.M., Wang, J.B., Hu, P. X., Nonlinear Schrödinger equation for internal waves in deep sea, Acta Physica Sinica, 59(02), 2010, 1123-1129
[14] Liu, S.K., Fu, Z.T., Expansion method about the Jacobi elliptic function and its applications to nonlinear wave equations, Acta Physica Sinica, 50, 2001, 2068-2073.
[15] He, J.H., Exp-function method for fractional differential equations, International Journal Nonlinear Science and Numerical Simulation, 14, 2013, 363-366.
[16] He, J.H., On the fractal variational principle for the Telegraph equation, Fractals, 29, 2021,
[17] Wu, Y., Variational approach to higher-order water-wave equations, Chaos Solitons and Fractals, 32, 2007, 195-203.
[18] Gazzola, F., Wang, Y., Pavani, R., Variational formulation of the Melan equation, Mathematical Methods in the Applied Sciences, 41, 2018, 943-951.
[19] He, J.H., Liu, F.J., Local Fractional Variational Iteration Method for Fractal Heat Transfer in Silk Cocoon Hierarchy, Nonlinear Science Letters A, 2013, 4, 15-20.
[20] He, J.H., Ji, F.Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57(8), 2019, 1932-1934.
[21] He, C.H., Shen, Y., Ji, F.Y., He, J.H., Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28(1), 2020, 2050011.
[22] He, J.H., Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Engineering Journal, 11(4), 2020, 1411-1414.
[23] Kaup, D.J., Variational solutions for the discrete nonlinear Schrödinger equation, Mathematics and Computers in Simulation, 69, 2005, 322-333.
[24] Putri, N.Z., Asfa, A.R., Fitri, A., Bakri, I., Syafwan, M., Variational approximations for intersite soliton in a cubic-quintic discrete nonlinear Schrödinger equation, Journal of Physics, Conference Series, 2019, 1317, 012-015.
[25] He, J.H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons and Fractals, 19, 2004, 847-851.
[26] He, J.H., A modified Li-He’s variational principle for plasma, International Journal Numerical Methods for Heat & Fluid Flow, 2019, doi,10.1108/HFF-06-2019-0523.
[27] He, J.H., Generalized equilibrium equations for shell derived from a generalized variational principle, Applied Mathematics Letters, 64, 2017, 94-100.
[28] He, J.H., Sun, C., A variational principle for a thin film equation, Journal of Mathematical Chemistry, 57, 2019, 2075-2081.
[29] 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, 23-28.
[30] He, J.H., Generalized Variational Principles for Buckling Analysis of Circular Cylinders, Acta Mechanica, 231, 2020, 899-906.
[31] He, J.H., A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals,2019,
[32] Yue, S., He, J.H., Variational principle for a generalized KdV equation in a fractal space, Fractals, 28(4), 2020, 2050069.
[33] Biswas, A., Zhou, Q., Ullah, M.Z., Triki, H., Moshokoa, S.P., Belic, M., Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle, Optik, 143, 2017, 131-134.
[34] Cao, X.Q., Guo Y.N., Hou, S.C., Zhang, C.Z., Peng, K.C., Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water, Symmetry, 12, 2020, 850.
[35] Cao, X.Q., Generalized variational principles for Boussinesq equation systems, Acta Physica Sinica, 60, 2011, 105-113.
[36] Kohl, R.W., Biswas, A., Zhou, Q., Ekici, M. Alzahrani, A.K., Belic, M.R., Optical soliton perturbation with polynomial and triple-power laws of refractive index by semi-inverse variational principle, Chaos, Solitons & Fractals, 135, 2020, 109765.
[37] He, J.H., El-Dib, Y.O., Periodic property of the time-fractional Kundu–Mukherjee–Naskar equation, Results in Physics, 19, 2020, 103345.
[38] He, J.H., Variational principle and periodic solution of the Kundu–Mukherjee–Naskar equation, Results in Physics, 17, 103031
[39] He, J.H., Kou, S.J., He, C.H., Zhang, Z.W., Khaled, A. Gepreel., Fractal oscillation and its frequency-amplitude property, Fractals, 2021, DOI: 10.1142/S0218348X2150105X.
[40] He, C.H., Liu, C., He, J.H., Shirazi, A.H., Sedighi, H.M., Passive Atmospheric water harvesting utilizing an ancient Chinese ink slab and its possible applications in modern architecture, Facta Universitatis Series-Mechanical Engineering, 2021, doi: 10.22190/FUME201203001H.