| Citation: |
Yu-Qi Chen, Bo Tian, Qi-Xing Qu, Cheng-Cheng Wei, Dan-Yu Yang. PAINLEVÉ INTEGRABLE PROPERTY, BILINEAR FORM, BÄCKLUND TRANSFORMATION, KINK AND SOLITON SOLUTIONS OF A (2+1)-DIMENSIONAL VARIABLE-COEFFICIENT GENERAL COMBINED FOURTH-ORDER SOLITON EQUATION IN A FLUID OR PLASMA[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 742-759. doi: 10.11948/20230056
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PAINLEVÉ INTEGRABLE PROPERTY, BILINEAR FORM, BÄCKLUND TRANSFORMATION, KINK AND SOLITON SOLUTIONS OF A (2+1)-DIMENSIONAL VARIABLE-COEFFICIENT GENERAL COMBINED FOURTH-ORDER SOLITON EQUATION IN A FLUID OR PLASMA
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Abstract
In this paper, we focus our attention on a (2+1)-dimensional variable-coefficient general combined fourth-order soliton equation in a fluid or plasma. Under certain coefficient constraints, we get the Painlevé integrable property. We obtain the bilinear form and bilinear auto-Bäcklund transformation. By virtue of the truncated Painlevé expansion, we derive an auto-Bäcklund transformation. Under certain coefficient constraints, we graphically analyse the one-kink waves, one soliton, two-kink waves and two solitons. We get the expressions of the amplitude and velocity of the one soliton and analyse the types of the two solitons and two-kink waves before and after the interactions.
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References
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Yu-Qi Chen, Bo Tian, Qi-Xing Qu, Cheng-Cheng Wei, Dan-Yu Yang. PAINLEVÉ INTEGRABLE PROPERTY, BILINEAR FORM, BÄCKLUND TRANSFORMATION, KINK AND SOLITON SOLUTIONS OF A (2+1)-DIMENSIONAL VARIABLE-COEFFICIENT GENERAL COMBINED FOURTH-ORDER SOLITON EQUATION IN A FLUID OR PLASMA[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 742-759. doi: 10.11948/20230056
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Figure 1.
One kink wave via Solutions (4.5) under Coefficient Constraints (4.4), (
$ a_1-c_1 $ $ y $ $ a_2-c_2 $ $ x $ $ a_1 $ $ a_2 $ $ \delta_2(t) $ $ t-1 $ $ \delta_3(t) $ $ -t $ $ \delta_4(t) $ $ \alpha $ -
Figure 2.
The same as Figs. 1 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_2(t) $ $ 2t^2-1 $ $ c_1 $ $ c_2 $ $ \delta_2(t) $ $ t(\sin t+t\cos t)-1 $ -
Figure 3.
The same as Figs. 1 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta _3(t) $ $ -0.5 $ $ c_1 $ $ c_2 $ $ \delta_3(t) $ $ -\frac{t}{\sin t+t\cos t} $ -
Figure 4.
The same as Figs. 1 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_4(t) $ $ 2t^2-1 $ $ c_1 $ $ c_2 $ $ \delta_4(t) $ $ t(\sin t+t\cos t)-1 $ -
Figure 5.
One soliton via Solutions (5.3) under Coefficient Constraints (5.2), (
$ a_1-c_1 $ $ y=0 $ $ a_2-c_2 $ $ x=0 $ $ a_1 $ $ a_2 $ $ \delta_2(t) $ $ t $ $ \delta_1(t) $ $ 1 $ $ \delta_4(t) $ $ 1 $ $ \alpha $ $ 1 $ -
Figure 6.
The same as Figs. 5 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta _1(t) $ $ t $ $ c_1 $ $ c_2 $ $ \delta_1(t) $ $ \sin t $ -
Figure 7.
The same as Figs. 5 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_2(t) $ $ 1 $ $ c_1 $ $ c_2 $ $ \delta_2(t) $ $ \sin t $ -
Figure 8.
The same as Figs. 5 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_4(t) $ $ t $ $ c_1 $ $ c_2 $ $ \delta_4(t) $ $ \sin t $ -
Figure 9.
Two solitons via Solutions (5.7) under Coefficient Constraints (5.6), (
$ a_1-c_1 $ $ y=0 $ $ a_2-c_2 $ $ x=0 $ $ a_1 $ $ a_2 $ $ \delta_4(t) $ $ 1 $ $ \alpha $ $ 1 $ $ \delta_1(t) $ $ 2 $ $ \delta_2(t) $ $ t $ -
Figure 10.
The same as Figs. 9 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_1(t) $ $ t $ $ \delta_2(t) $ $ 2t $ $ c_1 $ $ c_2 $ $ \delta_1(t) $ $ t $ $ \delta_2(t) $ $ \cos t $ -
Figure 11.
The same as Figs. 9 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_1(t) $ $ t $ $ \delta_4(t) $ $ 2t $ $ c_1 $ $ c_2 $ $ \delta_1(t) $ $ t $ $ \delta_4(t) $ $ \cos t $ -
Figure 12.
The same as Figs. 9 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_4(t) $ $ 2t $ $ c_1 $ $ c_2 $ $ \delta_4(t) $ $ \cos t $ -
Figure 13.
Two kink waves via Solutions (6.5) under Coefficient Constraints (6.4), (
$ a_1-c_1 $ $ y=0 $ $ a_2-c_2 $ $ x=0 $ $ a_1 $ $ a_2 $ $ \delta_4(t) $ $ 1 $ $ \alpha $ $ 1 $ $ \delta_1(t) $ $ 1 $ $ \delta_2(t) $ $ 3 $ -
Figure 14.
The same as Figs. 13 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_1(t) $ $ t $ $ \delta_2(t) $ $ 2 $ $ c_1 $ $ c_2 $ $ \delta_1(t) $ $ t $ $ \delta_2(t) $ $ \sin t $ -
Figure 15.
The same as Figs. 13 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_1(t) $ $ t $ $ \delta_4(t) $ $ 2 $ $ c_1 $ $ c_2 $ $ \delta_1(t) $ $ t $ $ \delta_4(t) $ $ \sin t $ -
Figure 16.
The same as Figs. 13 (
$ a_1 $ $ a_2 $ $ b_1 $ $ b_2 $ $ \delta_2(t) $ $ t $ $ \delta_4(t) $ $ 2 $ $ c_1 $ $ c_2 $ $ \delta_2(t) $ $ t $ $ \delta_4(t) $ $ \sin t $