| Citation: |
Jiangying Huo, Taogetusang Bao. BÄCKLUND TRANSFORMATION AND WEIERSTRASS ELLIPTIC FUNCTION SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL COUPLED DAVEY-STEWARTSON EQUATION IN SHALLOW WATER WAVES[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 413-431. doi: 10.11948/20250167
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BÄCKLUND TRANSFORMATION AND WEIERSTRASS ELLIPTIC FUNCTION SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL COUPLED DAVEY-STEWARTSON EQUATION IN SHALLOW WATER WAVES
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Abstract
This paper investigates the generalized (2+1)-dimensional coupled Davey-Stewartson (DS) equation, which describes the motions of long and short waves in shallow waters and has extensive applications in plasma physics. First, we study the Weierstrass elliptic function solutions using the auxiliary function method. Then, the bilinear Bäcklund transformation (BT) and Bell polynomial Bäcklund transformation are constructed using Bell polynomial method. Moreover, explicit 1-soliton, explicit 2-soliton, and explicit 3-soliton solutions are obtained from the derived bilinear form. Finally, the physical behaviors of the solutions are illustrated through three-dimensional plots, density plots, and two-dimensional planar graphs.
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References
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Jiangying Huo, Taogetusang Bao. BÄCKLUND TRANSFORMATION AND WEIERSTRASS ELLIPTIC FUNCTION SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL COUPLED DAVEY-STEWARTSON EQUATION IN SHALLOW WATER WAVES[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 413-431. doi: 10.11948/20250167
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Figure 1.
The propagation characteristics of expression
$ E_{1} $ $ \beta_{1}=\beta_{2}=\alpha_{1}=\alpha_{2}=\overline{C}=\lambda=1$ $c_{0}=0.5, l=4, \tau=5, w=3, \beta_{3}=2, \delta=-1 $ -
Figure 2.
The propagation characteristics of expression
$ E_{2} $ $ \beta_{1}=\beta_{2}=\alpha_{1}=\alpha_{2}=\overline{C}=\lambda=1$ $c_{0}=0.5, l=4, \tau=5, w=3, \beta_{3}=2, \delta=-1 $ -
Figure 3.
The propagation characteristics of expression
$ |E_{3}| $ $ \beta_{1}=\beta_{2}=\alpha_{1}=\alpha_{2}=\overline{C}=\lambda=1$ $c_{0}=0.5, l=4, \tau=5, w=3, \beta_{3}=2, \delta=-1 $ -
Figure 4.
The propagation characteristics of expression
$ E_{4} $ $ \beta_{1}=\beta_{2}=\alpha_{1}=\alpha_{2}=\delta=\overline{C}=\lambda=c_{0}=1, l=4, \tau=5, w=3, \beta_{3}=2 $ -
Figure 5.
The propagation characteristics of expression
$ E_{5} $ $ \beta_{1}=\beta_{2}=\alpha_{1}=\alpha_{2}=\delta=\overline{C}=\lambda=c_{0}=1, l=4, \tau=5, w=3, \beta_{3}=2 $ -
Figure 6.
The propagation characteristics of expression
$ E_{6} $ $ \beta_{1}=\beta_{2}=\alpha_{1}=\alpha_{2}=\delta=\overline{C}=\lambda=c_{0}=1, l=4, \tau=5, w=3, \beta_{3}=2 $ -
Figure 7.
Profiles of the explicit 1-soliton solution given by expression (4.7), with parameters:
$ \sigma_1=0.1, \sigma_2=0.2, \mu=0.2, \alpha_1=0.4, \alpha_2=1$ $\delta=1, s_1=4-\mathrm{i}, l_1=1+\mathrm{i}, \beta_1=2-\mathrm{i}$ $\beta_2=3+5\mathrm{i}, \beta_3=-0.3$ $\xi_1=3+2\mathrm{i}.(a, d) t=-10; (b, e) t=0; (c, f) t=10 $ -
Figure 8.
Profiles of the explicit 2-soliton solution given by expression (4.11), with parameters:
$ \sigma_1=0.1, \sigma_2=0.2, \mu=0.2, \alpha_1=0.4$ $\alpha_2=1, \delta=1, s_1=4-\mathrm{i}, l_1=1+\mathrm{i}, \beta_1=2-\mathrm{i}$ $\beta_2=3+5\mathrm{i}, \beta_3=-0.3, \xi_1=3+2\mathrm{i}$ $\xi_2=3-2\mathrm{i}.(a, d) t=-10; (b, e) t=0; (c, f) t=10 $ -
Figure 9.
Profiles of the explicit 3-soliton solution given by expression (4.15), with parameters:
$ \sigma_1=0.1, \sigma_2=0.2, \mu=0.2, \alpha_1=0.4$ $\alpha_2=1, \delta=1, s_1=4-\mathrm{i}, l_1=1+\mathrm{i}, \beta_1=2-\mathrm{i}$ $\beta_2=3+5\mathrm{i}, \beta_3=-0.3, \xi_1=3+2\mathrm{i}, \xi_2=3-2\mathrm{i}$ $\xi_3=2.(a, d) t=-20; (b, e) t=-10; (c, f) t=0.5 $