| Citation: |
Jianping Shi, Mengmeng Zhou, Hui Fang. GROUP-INVARIANT SOLUTIONS, NON-GROUP-INVARIANT SOLUTIONS AND CONSERVATION LAWS OF QIAO EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 2023-2036. doi: 10.11948/20190110
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GROUP-INVARIANT SOLUTIONS, NON-GROUP-INVARIANT SOLUTIONS AND CONSERVATION LAWS OF QIAO EQUATION
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Abstract
This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators. -
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References
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Jianping Shi, Mengmeng Zhou, Hui Fang. GROUP-INVARIANT SOLUTIONS, NON-GROUP-INVARIANT SOLUTIONS AND CONSERVATION LAWS OF QIAO EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 2023-2036. doi: 10.11948/20190110
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Figure 1. A group-invariant solution
$ u(t, x) = \frac{c_{1}\exp(x-\frac{\beta\ln(2\gamma t+\alpha)}{2\gamma}))+c_{2}\exp(-x+\frac{\beta\ln(2\gamma t+\alpha)}{2\gamma}))}{\sqrt{2\gamma t+\alpha}} $ with$ c_1 = 0.05, c_2 = 0.2, \alpha = 2, \beta = 1, \gamma = 1. $ -
Figure 2. Two traveling wave-like solutions with variable amplitude
$ \hat{u}(t, x) = g(t)(c_{1}\exp(x-ct)+c_{2}\exp(-x+ct)) $ with$ c = 0.01, c_1 = c_2 = 0.05, $ and$ g(t) = t+\sin^2 t $ in (a),$ g(t) = 0.5+0.15\sin(2t) $ in (b) -
Figure 3. Two traveling wave-like solutions with variable velocity
$ \hat{u}(t, x) = g(t)(c_{1}\exp(x-c(t)t)+c_{2}\exp(-x+c(t)t)) $ with$ c_1 = c_2 = 0.05, c(t) = 0.001+\sin^2 t, $ and$ g(t) = 1 $ in (a),$ g(t) = 0.2+0.05\sin t $ in (b) -
Figure 4. Two traveling wave-like solutions with variable amplitude and variable velocity
$ \hat{u}(t, x) = \frac{h(t)}{\sqrt{2\gamma t+\alpha}}(c_{1}\exp(x-\frac{\beta\ln(2\gamma t+\alpha)}{2\gamma})+c_{2}\exp(-x+\frac{\beta\ln(2\gamma t+\alpha)}{2\gamma})) $ with$ c_1 = 0.05, c_2 = 0.2, \alpha = 2, \beta = 1, \gamma = 1, $ and$ h(t) = 1+\sin^2 t $ in (a),$ h(t) = \exp(\frac{t}{10}) $ in (b)