| Citation: |
Shuangshuang Xia, Zenggui Wang. LIE SYMMETRIES, GROUP INVARIANT SOLUTIONS AND CONSERVATION LAWS OF IDEAL MHD EQUATIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1959-1986. doi: 10.11948/20210410
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LIE SYMMETRIES, GROUP INVARIANT SOLUTIONS AND CONSERVATION LAWS OF IDEAL MHD EQUATIONS
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Abstract
In this paper, the method of Lie symmetry is used to study the incompressible ideal MHD equations. We derive the infinitesimal generators of MHD by using Lie symmetry method. Then, we get the group invariant solutions to MHD based on the group transformation and known solutions. An one-dimensional optimal system which only depends on the commutator of Lie algebras is constructed. Through the similarity transformation of the optimal system, we obtain the 1 + 1-dimensional reduced partial differential equations, and further obtain the exact solutions of these equations, such as Bell soliton solution, junction soliton solution and seed solution. In addition, we prove the nonlinear self-adjoint and conservation laws of the MHD equations using the Ibragimov method. Finally, the classical symmetry and solutions to incompressible ideal MHD equations with initial conditions left invariant are discussed.
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References
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Shuangshuang Xia, Zenggui Wang. LIE SYMMETRIES, GROUP INVARIANT SOLUTIONS AND CONSERVATION LAWS OF IDEAL MHD EQUATIONS[J]. Journal of Applied Analysis & Computation, 2022, 12(5): 1959-1986. doi: 10.11948/20210410
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- Figure 1. The solution u with y=1: (a). indicates the solitons solution; (b).indicates the density plot of u; (c). indicates the wave track along the t-axis at t=1, t=5 and along the x-axis at x=0, x=10.
- Figure 2. The solution v with y=1 : (a). indicates the solitons solution; (b).indicates the density plot of v; (c). indicates the wave track along the t-axis at t=1, t=4 and along the x-axis at x=-10, x=10.
- Figure 3. The solution p with y=0: (a). indicates the solitons solution; (b).indicates the density plot of p; (c). indicates the wave track along the t-axis at t=±1, t=±4.
- Figure 4. The solution u: (a). indicates the multiple soliton solution; (b).indicates the density plot of u; (c). indicates the wave track along the x-axis at x=0, x=2, x=5 and along the y-axis at x=0, y=5.
- Figure 5. The solution b: (a). indicates the bright soliton solution; (b).indicates the density plot of u; (c). indicates the wave track along the x-axis at x=0, x=±5.
- Figure 6. The solution u: (a). indicates the multiple soliton solution; (b).indicates the density plot of u; (c). indicates the wave track along the x-axis at x=0, x=2, x=5.
- Figure 7. The solution b: (a). indicates the break multiple soliton solution; (b).indicates the density plot of u; (c). indicates the wave track along the x-axis at x=0, x=±5.
- Figure 8. The solution u: (a). indicates the periodic soliton solution; (b).indicates the density plot of u; (c). indicates the wave track along the x-axis at x=2, x=5 and along the t-axis at t=2, t=10.
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Figure 9. The solution v: (a). indicates the bell soliton solution; (b).indicates the density plot of v; (c). indicates the wave track along the x-axis at x=±4 and along the t-axis at
$ t=\frac{1}{2} $ , t=1.