| Citation: |
Jian-Guo Liu, Abdul-Majid Wazwaz, Wen-Hui Zhu. SOLITARY AND LUMP WAVES INTERACTION IN VARIABLE-COEFFICIENT NONLINEAR EVOLUTION EQUATION BY A MODIFIED ANSÄTZ WITH VARIABLE COEFFICIENTS[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 517-532. doi: 10.11948/20210178
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SOLITARY AND LUMP WAVES INTERACTION IN VARIABLE-COEFFICIENT NONLINEAR EVOLUTION EQUATION BY A MODIFIED ANSÄTZ WITH VARIABLE COEFFICIENTS
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Abstract
In this work, we examine variable-coefficient nonlinear evolution equations that often describe complex physical models more than constant coefficient models. A modified ansätz with variable coefficients is used for studying the solitary and lump waves interaction in these variable-coefficient nonlinear evolution equations. We discuss the variable-coefficient Kadomtsev-Petviashvili equation to achieve this goal. We present lump wave and interaction solutions between solitary and lump waves for this model. By choosing appropriate values of the variable coefficients, 3d plots and corresponding contour plots are drawn to illustrate the dynamical behaviors of the obtained solutions.
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References
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Jian-Guo Liu, Abdul-Majid Wazwaz, Wen-Hui Zhu. SOLITARY AND LUMP WAVES INTERACTION IN VARIABLE-COEFFICIENT NONLINEAR EVOLUTION EQUATION BY A MODIFIED ANSÄTZ WITH VARIABLE COEFFICIENTS[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 517-532. doi: 10.11948/20210178
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Figure 1. Lump wave with
$ \phi(t) = \mu_1 = \Psi _0 = \Psi _5(t) = 1 $ ,$ \Psi _1(t) = 2 $ ,$ y = 0 $ ,$ \Psi _2(t) = \Psi _4(t) = -1 $ , (a) 3d plot; (b) corresponding contour plot. -
Figure 2. Lump wave with
$ \phi(t) = t $ ,$ \mu_1 = \Psi _0 = \Psi _5(t) = 1 $ ,$ \Psi _1(t) = 2 $ ,$ y = 0 $ ,$ \Psi _2(t) = \Psi _4(t) = -1 $ , (a) 3d plot; (b) corresponding contour plot. -
Figure 3. Lump wave with
$ \phi(t) = 1+\sin t $ ,$ \mu_1 = \Psi _0 = \Psi _5(t) = 1 $ ,$ \Psi _1(t) = 2 $ ,$ y = 0 $ ,$ \Psi _2(t) = \Psi _4(t) = -1 $ , (a) 3d plot; (b) corresponding contour plot. -
Figure 4. Interaction solutions between lump wave and one solitary wave with
$ \phi(t) = \mu_4 = \Psi _0 = 1 $ ,$ \Psi _1(t) = 2 $ ,$ \Psi _2(t) = \mu_3 = \mu_2 = -1 $ ,$ \varsigma _3(t) = t $ , when$ y = -5 $ in (a) (d),$ y = 0 $ in (b) (e) and$ y = 5 $ in (c) (f). -
Figure 5. Interaction solutions between lump wave and one solitary wave with
$ \phi(t) = \cos (2 t)+1 $ ,$ \mu_4 = \Psi _0 = 1 $ ,$ \Psi _1(t) = 2 $ ,$ \Psi _2(t) = \mu_3 = \mu_2 = -1 $ ,$ \varsigma _3(t) = t $ , when$ y = -5 $ in (a) (d),$ y = 0 $ in (b) (e) and$ y = 5 $ in (c) (f). -
Figure 6. Interaction solutions between lump wave and two solitary waves with
$ \phi(t) = \mu_6 = \mu_5 = \Psi _0 = 1 $ ,$ \Psi _1(t) = 2 $ ,$ \Psi _2(t) = -1 $ ,$ \varsigma _3(t) = t $ when$ y = -5 $ in (a) (d),$ y = 0 $ in (b) (e) and$ y = 5 $ in (c) (f). -
Figure 7. Interaction solutions between lump wave and two solitary waves with
$ \phi(t) = \cos (2 t)+1 $ ,$ \mu_6 = \mu_5 = \Psi _0 = 1 $ ,$ \Psi _1(t) = 2 $ ,$ \Psi _2(t) = -1 $ ,$ \varsigma _3(t) = t $ , when$ y = -5 $ in (a) (d),$ y = 0 $ in (b) (e) and$ y = 5 $ in (c) (f). -
Figure 8. lump wave (3.3) when
$ x = 0 $ in (a),$ y = 0 $ in (b) and$ t = 0 $ in (c). -
Figure 9. lump wave (2.4) with
$ y = 0 $ when$ \phi(t) = 1- 12 \sin t $ in (a),$ \phi(t) = 1+4 \sin t $ in (b) and$ \phi(t) = 1-28 \sin t $ in (c).