| Citation: |
Penghong Zhong, Ye Chen, Song Chen, Ganshan Yang. SOLITONS AND DOMAIN-WALL-ARRAY SOLUTIONS OF THE SCHRÖDINGER FLOW AND LANDAU-LIFSHITZ EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1388-1420. doi: 10.11948/20220200
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SOLITONS AND DOMAIN-WALL-ARRAY SOLUTIONS OF THE SCHRÖDINGER FLOW AND LANDAU-LIFSHITZ EQUATION
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Abstract
We obtained some solitons and domain-wall-array solutions of the multidimensional Schrödinger flow and the Landau-Lifshitz equation using the homogeneous balance principle and general Jacobi elliptic-function method. These solutions include bright solitons and periodic solutions in terms of elliptic functions. We excluded several special types of solutions, such as kink profile solutions and dark solitons. The total phase profile of the solitons have two components: the kinematic origin, and the self-steepening effect. For the domain-wall-array solutions, the total phase profile consists of the kinematic origin, kinematic chirping, and self-steepening effect. In certain parameter domains, fundamental domain wall-array-solutions are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. For ODEs deduced from Schrödinger flow that have no analytical solution, phase analysis is used to identify and classify the typical evolutionary pattern. Furthermore, the existence of limit cycles is verified, and the locations of singularities are precisely estimated.
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Keywords:
- Schrödinger flow /
- Landau-Lifshitz equation /
- soliton /
- domain wall
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References
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Penghong Zhong, Ye Chen, Song Chen, Ganshan Yang. SOLITONS AND DOMAIN-WALL-ARRAY SOLUTIONS OF THE SCHRÖDINGER FLOW AND LANDAU-LIFSHITZ EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1388-1420. doi: 10.11948/20220200
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Figure 1.
The direction field of (4.2) with different values of
$ k $ $ w $ -
Figure 2.
The direction field of (4.2) with different values of
$ k $ $ w $ -
Figure 3.
The direction field of (4.5) with different values of
$ k $ $ w $ -
Figure 4.
The direction field of (4.5) with different values of
$ k $ $ w $ -
Figure 5.
(a)
$ |u| $ $ \omega=-3/2 $ $ \xi_0=0 $ $ \omega=-3/2 $ $ \xi_0=-2 $ $ \omega=-5/4 $ $ \xi_0=0 $ $ \omega=-5/4 $ $ \xi_0=2 $ -
Figure 6.
(a) The solution
$ |u| $ $ g(\xi)= {{{\rm{\,dn}}}}(\xi , R) $ $ R=9/10 $ $ R=99/100 $ $ R=1 $ $ 2-1/2\,{k}^{-2}+{g}^{-2}-1/2\,{{{k}^{-2}{g}^{-2}}}+{g}^{2} $ $ g(\xi)={{{\rm{\,sn}}}}(\xi , R) $ $ R=1/2 $ $ R=2/3 $ $ R=3/4 $ $ R=4/5 $ -
Figure 7.
Phase with the chirping reversal.
$ R=1/2 $ $ R=2/3 $ $ R=3/4 $ $ R=4/5 $ -
Figure 8.
(a) Contours of the real part of the bright soliton (2.13)-(2.14) depend on
$ t $ $ \xi $ $ t $ $ X $ $ \xi=X-\frac{t}{2} $ -
Figure 9.
(a) Contours of the real part of (3.5) with
$ g(\xi)={{{\rm{\,dn}}}}(\xi , R) $ $ g(\xi)= {{{\rm{\,cn}}}}(\xi , R) $ -
Figure 10.
(a) Figure of the solution 2.13 in terms of
$ \xi $ $ x $ $ \xi=x-\frac{t}{2} $ $ \xi=-\frac{\pi}{4} $ $ x $ -
Figure 11.
(a) The real part of (3.5) with
$ g(\xi)= {{{\rm{\,dn}}}}(\xi , R) $ $ g(\xi)= {{{\rm{\,cn}}}}(\xi , R) $ -
Figure 12.
(a) The real part of (3.5) with
$ g(\xi)= {{{\rm{\,nc}}}}(\xi , R) $ $ |u| $ $ g(\xi)= {{{\rm{\,nc}}}}(\xi , R) $ -
Figure 13.
(a) The real part of (3.5) with
$ g(\xi)= {{{\rm{\,nd}}}}(\xi , R) $ $ |u| $ $ g(\xi)= {{{\rm{\,nd}}}}(\xi , R) $ -
Figure 14.
(a) The real part of (3.5) with
$ g(\xi) $ $ |u| $ $ g(\xi) $ -
Figure 15.
(a) Solution (2.16) with
$ g(\xi) $ $ g(\xi) $ -
Figure 16.
(a) Solution 3.6 with with
$ g(\xi) $ $ g(\xi) $ $ g(\xi) $ $ g(\xi) $