2021 Volume 11 Issue 2
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Lei Fu, Huanhe Dong, Chaudry Masood Khalique, Hongwei Yang. CONSERVATION LAWS, EXACT SOLUTIONS OF TIME-SPACE FRACTIONAL GENERALIZED GINZBURG-LANDAU EQUATION FOR SHALLOW WAKE FLOWS[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 874-891. doi: 10.11948/20200053
Citation: Lei Fu, Huanhe Dong, Chaudry Masood Khalique, Hongwei Yang. CONSERVATION LAWS, EXACT SOLUTIONS OF TIME-SPACE FRACTIONAL GENERALIZED GINZBURG-LANDAU EQUATION FOR SHALLOW WAKE FLOWS[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 874-891. doi: 10.11948/20200053
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CONSERVATION LAWS, EXACT SOLUTIONS OF TIME-SPACE FRACTIONAL GENERALIZED GINZBURG-LANDAU EQUATION FOR SHALLOW WAKE FLOWS

  • 1.

    College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590

  • 2.

    Arican Institute for Mathematical and Science, 6 Meirose Road, Muizenberg Cape Town 7945, South Africa

  • 3.

    International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafkeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

    Abstract

  • In this study, starting from the rigid-lid shallow water equations, an amplitude evolution equation is derived by using the multi-scale and perturbation analysis method. The resulting equation has complex coefficients and is called (2+1)-dimensional generalized Ginzburg-Landau(gGL) equation. Then, the (2+1)-dimensional time-space fractional gGL equation is obtained by using the semi-inverse method and fractional variational principle in the first time. Finally, the conservation laws and exact solutions of the fractional gGL equation are discussed on the basis of Lie symmetry analysis and $ {\rm exp}(-\phi(\zeta)) $-expansion method. By analyzing these solutions, we conclude that there are solitary waves and rogue waves in shallow wake flows.
    MSC: 34A08, 76D25, 35Q56, 35L65
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