| Citation: |
Qing Meng, Bin He. BIFURCATION ANALYSIS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERIC TWO-DIMENSIONAL SINE-GORDON EQUATION IN NONLINEAR OPTICS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1443-1463. doi: 10.11948/20190227
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BIFURCATION ANALYSIS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERIC TWO-DIMENSIONAL SINE-GORDON EQUATION IN NONLINEAR OPTICS
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Abstract
We focus on investigating a generic two-dimensional sine-Gordon equation in nonlinear optics. Based on a viable transformation, the bifurcation analysis of the equation is carried out in this paper. The phase portraits are given and different kinds of traveling wave solutions are obtained. The analytical results are also numerically simulated. -
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References
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Qing Meng, Bin He. BIFURCATION ANALYSIS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERIC TWO-DIMENSIONAL SINE-GORDON EQUATION IN NONLINEAR OPTICS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1443-1463. doi: 10.11948/20190227
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Figure 1. Bifurcations and phase portraits of system (2.7) when
$ a^2-bc<0,\alpha<0. $ Parameters: (a)$ g<0,\beta<0. $ (b)$ g<0,\beta\geq0. $ (c)$ g = 0,\beta>0. $ (d)$ g = 0,\beta = 0. $ (e)$ g = 0,\beta<0. $ (f)$ g>0,\beta\leq0. $ (g)$ g>0,0<\beta<-\frac{1}{4\alpha}g^2. $ (h)$ g>0,\beta = -\frac{1}{4\alpha}g^2. $ (i)$ g>0,\beta>-\frac{1}{4\alpha}g^2. $ -
Figure 2. Bifurcations and phase portraits of system (2.7) when
$ a^2-bc<0,\alpha = 0. $ Parameters: (a)$ g<0,\beta<0. $ (b)$ g<0,\beta>0. $ (c)$ g = 0,\beta>0. $ (d)$ g = 0,\beta<0. $ (e)$ g>0,\beta<0. $ (f)$ g>0,\beta>0. $ -
Figure 3. Bifurcations and phase portraits of system (2.7) when
$ a^2-bc<0,\alpha>0. $ Parameters: (a)$ g<0,\beta<-\frac{1}{4\alpha}g^2. $ (b)$ g<0,\beta = -\frac{1}{4\alpha}g^2. $ (c)$ g<0,-\frac{1}{4\alpha}g^2<\beta<0. $ (d)$ g<0,\beta\geq0. $ (e)$ g = 0,\beta>0. $ (f)$ g = 0,\beta = 0. $ (g)$ g = 0,\beta<0. $ (h)$ g>0,\beta\leq0. $ (i)$ g>0,\beta>0. $ - Figure 4. The level curves given by $H(u, y)=h$ when $h>0.$ Parameters:(a) $a^2-bc < 0, \alpha < 0, g=0, \beta=0.$(b) $a^2-bc < 0$, $\alpha$>0, g=0, $\beta$=0.
- Figure 5. Solitary waves of (2.6). Parameters: (a) $a^2-bc$=-1.1, α=-1.5, h=1.2. (b) $a^2-bc$=-1.1, α=-1.5, h=4.5. (c) $a^2-bc$=-1.1, α=1.5, h=1.2. (d) $a^2-bc$=-1.1, α=1.5, h=4.5. (e) $a^2-bc$=-2.5, α=-1.2, β=-0.5. (f) $a^2-bc$=-2.5, α=1.2, β=0.5.
- Figure 6. Blow-up waves of (2.6). Parameters: (a) $a^2-bc$=-1.1, α=-1.5, h=1.2. (b) $a^2-bc$=-1.1, α=-1.5, h=4.5. (c) $a^2-bc$=-1.1, α=1.5, h=1.2. (d) $a^2-bc$=-1.1, α=1.5, h=4.5. (e) $a^2-bc$=-1.1, α=-1.5. (f) $a^2-bc$=-1.1, α=1.5. (g) $a^2-bc$=-2.5, α=-1.2, β=-0.5. (h) $a^2-bc$=-2.5, α=1.2, β=0.5. (i) $a^2-bc$=-0.15, β=1.25. (j) $a^2-bc$=-0.15, β=-1.25. (k) $a^2-bc$=-3.0, β=1.75, h=1.25. (l) $a^2-bc$=-3.0, β=-1.75, h=1.25. (m) $a^2-bc$=-3.0, β=1.75, h=1.25. (n) $a^2-bc$=-3.0, β=-1.75, h=1.25.
- Figure 7. The level curves given by $H(u, y)=0.$ Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β=0. (b) $a^2-bc$ < 0, α>0, g=0, β=0.
- Figure 8. The level curves given by $H(u, y)=h$ when $h < 0.$ Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β=0. (b) $a^2-bc$ < 0, α>0, g=0, β=0.
- Figure 9. Periodic blow-up waves of (2.6). Parameters: (a) $a^2-bc$=-1.1, α=-1.5, h=-1.2. (b) $a^2-bc$=-1.1, α=-1.5, h=-4.5. (c) $a^2-bc$=-1.1, α=1.5, h=-1.2. (d) $a^2-bc$=-1.1, α=1.5, h=-4.5. (e) $a^2-bc$=-2.5, α=-1.2, β=-0.5, h=-1.5. (f) $a^2-bc$=-2.5, α=-1.2, β=-0.5, h=-3.5. (g) $a^2-bc$=-2.5, α=1.2, β=0.5, h=-1.5. (h) $a^2-bc$=-2.5, α=1.2, β=0.5, h=-1.5. (i) $a^2-bc$=-1.2, α=-1.0, β=-2.0. (j) $a^2-bc$=-1.2, α=-1.0, β=2.0. (k) $a^2-bc$=-1.5, α=-2.0, β=-1.2, h=4.5. (l) $a^2-bc$=-1.5, α=-2.0, β=-1.2, h=8.0. (m) $a^2-bc$=-1.5, α=2.0, β=1.2, h=4.5. (n) $a^2-bc$=-1.5, α=2.0, β=1.2, h=8.0. (o) $a^2-bc$=-2.5, α=-1.2, β=-0.5, h=-0.6. (p) $a^2-bc$=-2.5, α=-1.2, β=-0.5, h=0.6. (q) $a^2-bc$=-2.5, α=1.2, β=0.5, h=-0.6. (r) $a^2-bc$=-2.5, α=1.2, β=0.5, h=0.6. (s) $a^2-bc$=-0.1, α=-1.0, β=2.2, h=-2.0. (t) $a^2-bc$=-0.1, α=-1.0, β=2.2, h=2.0. (u) $a^2-bc$=-0.1, α=1.0, β=-2.2, h=-2.0. (v) $a^2-bc$=-0.1, α=1.0, β=-2.2, h=2.0.
- Figure 10. The level curves given by $H(u, y)=\frac{4\sqrt{αβ}}{a^2-bc}$. Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β < 0. (b) $a^2-bc$ < 0, α>0, g=0, β>0.
- Figure 11. The level curves given by $H(u, y)=h$ when $h < \frac{4\sqrt{αβ}}{a^2-bc}$. Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β < 0. (b) $a^2-bc$ < 0, α>0, g=0, β>0.
- Figure 12. Smooth periodic waves of (2.6). Parameters: (a) $a^2-bc$=-2.5, α=-1.2, β=-0.5, h=-1.5. (b) $a^2-bc$=-2.5, α=-1.2, β=-0.5, h=-3.5. (c) $a^2-bc$=-2.5, α=1.2, β=0.5, h=-1.5. (d) $a^2-bc$=-2.5, α=1.2, β=0.5, h=-1.5. (e) $a^2-bc$=-1.5, α=-2.0, β=-1.2, h=4.5. (f) $a^2-bc$=-1.5, α=-2.0, β=-1.2, h=8.0. (g) $a^2-bc$=-1.5, α=2.0, β=1.2, h=4.5. (h) $a^2-bc$=-1.5, α=2.0, β=1.2, h=8.0. (i) $a^2-bc$=-0.1, α=-1.0, β=2.2, h=-2.0. (j) $a^2-bc$=-0.1, α=-1.0, β=2.2, h=2.0. (k) $a^2-bc$=-0.1, α=1.0, β=-2.2, h=-2.0. (l) $a^2-bc$=-0.1, α=1.0, β=-2.2, h=2.0. (m) $a^2-bc$=-0.2, β=3.0, h=-0.5. (n) $a^2-bc$=-0.2, β=3.0, h=-2.0. (o) $a^2-bc$=-0.2, β=-3.0, h=-0.5. (p) $a^2-bc$=-0.2, β=-3.0, h=-2.0.
- Figure 13. The level curves given by $H(u, y)=-\frac{4\sqrt{αβ}}{a^2-bc}.$ Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β < 0. (b) $a^2-bc$ < 0, α>0, g=0, β>0.
- Figure 14. The level curves given by $H(u, y)=h$ when $h>-\frac{4\sqrt{αβ}}{a^2-bc}$. Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β < 0. (b) $a^2-bc$ < 0, α>0, g=0, β>0.
- Figure 15. The level curves given by $H(u, y)=h$ when $frac{4\sqrt{αβ}}{a^2-bc} < h < -\frac{4\sqrt{αβ}}{a^2-bc}$. Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β < 0. (b) $a^2-bc$ < 0, α>0, g=0, β>0.
- Figure 16. The level curves given by $H(u, y)=h$ when $h\in \mathbb{R}.$ Parameters: (a) $a^2-bc$ < 0, α < 0, g=0, β>0. (b) $a^2-bc$ < 0, α>0, g=0, β < 0.
- Figure 17. The level curves given by $H(u, y)=h$ when $h < 0.$ Parameters: (a) $a^2-bc$ < 0, α=0, g=0, β>0. (b) $a^2-bc$ < 0, α=0, g=0, β < 0.
- Figure 18. The level curves given by $H(u, y)=0$ Parameters: (a) $a^2-bc$ < 0, α=0, g=0, β>0. (b) $a^2-bc$ < 0, α=0, g=0, β < 0.
- Figure 19. The level curves given by $H(u, y)=h$ when $h>0.$ Parameters: (a) $a^2-bc$ < 0, α=0, g=0, β>0. (b) $a^2-bc$ < 0, α=0, g=0, β<0.