Communications in Theoretical Physics ›› 2021, Vol. 73 ›› Issue (2): 025005. doi: 10.1088/1572-9494/abd0e5
• Mathematical Physics • Previous Articles Next Articles
Tao Xu1,2(
),Guowei Zhang2,Liqun Wang2,Xiangmin Xu2,Min Li3,
Received: 2020-08-22
Revised: 2020-10-13
Accepted: 2020-12-06
Published: 2021-02-01
Contact:
Tao Xu
E-mail:xutao@cup.edu.cn
Tao Xu,Guowei Zhang,Liqun Wang,Xiangmin Xu,Min Li, Commun. Theor. Phys. 73 (2021) 025005.
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Figure 1.
Three types of one-soliton solutions of equation (2): (a) Aan SH soliton with β1=0, γ1=1 and ${\lambda }_{1}=\tfrac{3}{5}+\tfrac{2}{5}{\rm{i}};$ (b) a DH soliton with β1=1, γ1=0 and ${\lambda }_{1}=\tfrac{3}{5}+\tfrac{1}{10}{\rm{i}};$ (c) an MH soliton with β1=0.5, γ1=1 and ${\lambda }_{1}=\tfrac{1}{2}+\tfrac{1}{5}{\rm{i}}$."
Figure 2.
Time evolution for three different one-soliton solutions of equation (2): (a) an SH soliton with β1=0, γ1=2 and λ1=−0.25 + 0.2i; (b) a DH soliton with β1=2, γ1=0 and λ1=−0.3 + 0.06i; (c) an MH soliton with β1=0.5, γ1=2 and ${\lambda }_{1}=-0.2+\tfrac{\pi }{16}\,{\rm{i}}$."
Figure 3.
Time evolution of error norms for three cases of one-soliton solutions in figures 2(a)-(c)."
Figure 4.
Time evolution of discrete masses for three cases of one-soliton solutions in figures 2(a)-(c)."
Figure 5.
Time evolution of three different two-soliton solutions of equation (2): (a) shape-preserving collision between an SH soliton and a DH soliton with β1=β2=1, γ1=γ2=0, λ1=−0.32 − 0.1i and λ2=0.32 − 0.3i; (b) shape-preserving collision between two MH solitons with β1=β2=1, γ1=γ2=1, λ1=−0.375 − 0.12i and λ2=0.375 − 0.3i; (c) shape-changing collision between an MH soliton and a DH soliton with β1=β2=1, γ1=0, γ2=1, λ1=0.4 − 0.08i and λ2=−0.3 − 0.3i."
Figure 6.
Transverse plots of the two-soliton collision associated with figure 6(a) at (a) ξ=0, (b) ξ=17, (c) ξ=23, and (d) ξ=40."
Figure 7.
Time evolution of error norms for three cases of two-soliton solutions in figures 5(a)-(c)."
Figure 8.
Time evolution of discrete masses for three cases of two-soliton solutions in figures 5(a)-(c)."
Table 1.
Spatial error analysis of three one-soliton solutions with time step τ=0.01 and different mesh sizes."
| Mesh size | e∞ (SH soliton) | Order | e∞ (DH soliton) | Order | e∞ (MH soliton) | Order |
|---|---|---|---|---|---|---|
| h = 0.1 | 1.36E-3 | — | 5.98E-4 | — | 4.26E-3 | — |
| $h=\tfrac{0.1}{2}$ | 3.41E-4 | 1.9977 | 1.50E-5 | 1.9966 | 1.06E-3 | 2.0030 |
| $h=\tfrac{0.1}{4}$ | 8.55E-5 | 1.9957 | 3.76E-5 | 1.9934 | 2.69E-4 | 1.9835 |
| $h=\tfrac{0.1}{8}$ | 2.17E-5 | 1.9784 | 9.61E-6 | 1.9747 | 6.85E-5 | 1.9742 |
| $h=\tfrac{0.1}{16}$ | 5.78E-6 | 1.9083 | 2.90E-6 | 1.8750 | 1.80E-5 | 1.9298 |
Table 2.
Time error analysis of three one-soliton solutions with mesh step h = 0.001 and different time steps."
| Time step | e∞ (SH soliton) | Order | e∞ (DH soliton) | Order | e∞ (MH soliton) | Order |
|---|---|---|---|---|---|---|
| τ=0.1 | 7.41E-5 | — | 8.36E-5 | — | 5.70E-4 | — |
| $\tau =\tfrac{0.1}{2}$ | 1.86E-5 | 1.9977 | 2.10E-5 | 1.9989 | 1.43E-4 | 1.9944 |
| $\tau =\tfrac{0.1}{4}$ | 4.68E-6 | 1.9957 | 5.26E-6 | 1.9921 | 3.59E-5 | 1.9921 |
| $\tau =\tfrac{0.1}{8}$ | 1.26E-6 | 1.9784 | 1.34E-6 | 1.9921 | 9.08E-6 | 1.9852 |
| $\tau =\tfrac{0.1}{16}$ | 5.94E-7 | 1.9083 | 3.70E-7 | 1.7270 | 2.42E-6 | 1.9063 |
Figure 9.
Time evolution of the error norms from ξ=0 to 10 for three types of one-soliton solutions with different initial perturbations, where the parameters are chosen as follows: (a) β1=0, γ1=2 and λ1=− 0.25 + 0.2i, (b) β1=2, γ1=0 and λ1=−0.3 + 0.06i, (c) β1=0.5, γ1=2 and ${\lambda }_{1}=-0.2+\tfrac{\pi }{16}\,{\rm{i}}$."
Figure 10.
Comparison of the exact analytical solution and numerical solutions at ξ=10 for (a) SH soliton, (b) DH soliton, and (c) MH soliton, where $\varepsilon =\tfrac{1}{100}$ and the other parameters are the same as those in figure 9."
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