• Previous Article
    Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case
  • ERA Home
  • This Issue
  • Next Article
    Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra
doi: 10.3934/era.2021002

Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system

1. 

Department of Mathematics, Faculty of Sciences, University of Tabuk, Saudi Arabia, Lab. of Algebra, Number Theory and Nonlinear Analysis, Department of Mathematics

2. 

Faculty of Sciences, University of Monastir, 5000 Monastir, Tunisia

3. 

Department of Mathematics, Faculty of Sciences, University of Tabuk, Saudi Arabia

4. 

Department of Mathematics, Higher Institute of Applied Mathematics and Computer, Science, University of Kairouan, Street of Assad Ibn Alfourat, 3100 Kairouan, Tunisia, Lab. of Algebra, Number Theory and Nonlinear Analysis, Department of Mathematics

5. 

Department of Mathematics, Faculty of Sciences, University of Tabuk, Saudi Arabia

* Corresponding author: Anouar Ben Mabrouk

Received  August 2020 Revised  November 2020 Published  January 2021

In this paper, we study a couple of NLS equations characterized by mixed cubic and super-linear sub-cubic power laws. Classification as well as existence and uniqueness of the steady state solutions have been investigated. Numerical simulations have been also provided illustrating graphically the theoretical results. Such simulations showed that possible chaotic behaviour seems to occur and needs more investigations.

Citation: Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, doi: 10.3934/era.2021002
References:
[1]

J. S. AitchisonA. M. WeinerY. SilberbergM. K. OliverJ. L. JackelD. E. LeairdE. M. Vogel and P. W. E. Smith, Observation of spatial optical solitons in a nonlinear glass waveguide, Opt. Lett., 15 (1990), 471-473.  doi: 10.1364/OL.15.000471.  Google Scholar

[2]

H. AminikhahF. Pournasiri and F. Mehrdoust, A novel effective approach for systems of coupled Schrödinger equation, Pramana, 86 (2016), 19-30.  doi: 10.1007/s12043-015-0961-4.  Google Scholar

[3]

B. BalabaneJ. Dolbeault and H. Ounaeis, Nodal solutions for a sublinear elliptic equation, Nonlinear Anal., 52 (2003), 219-237.  doi: 10.1016/S0362-546X(02)00104-9.  Google Scholar

[4]

A. Ben Mabrouk and M. Ayadi, A linearized finite-difference method for the solution of some mixed concave and convex nonlinear problems, Appl. Math. Comput., 197 (2008), 1-10.  doi: 10.1016/j.amc.2007.07.051.  Google Scholar

[5]

A. Ben Mabrouk and M. Ayadi, Lyapunov type operators for numerical solutions of PDEs, Appl. Math. Comput., 204 (2008), 395-407.  doi: 10.1016/j.amc.2008.06.061.  Google Scholar

[6]

A. Ben Mabrouk and M. L. Ben Mohamed, Nodal solutions for some nonlinear elliptic equations, Appl. Math. Comput., 186 (2007), 589-597.  doi: 10.1016/j.amc.2006.08.003.  Google Scholar

[7]

A. Ben Mabrouk and M. L. Ben Mohamed, Phase plane analysis and classification of solutions of a mixed sublinear-superlinear elliptic problem, Nonlinear Anal., 70 (2009), 1-15.  doi: 10.1016/j.na.2007.11.041.  Google Scholar

[8]

A. Ben Mabrouk and M. L. Ben Mohamed, Nonradial solutions of a mixed concave-convex elliptic problem, J. Partial Differ. Equ., 24 (2011), 313-323.  doi: 10.4208/jpde.v24.n4.3.  Google Scholar

[9]

A. Ben Mabrouk and M. L. Ben Mohamed, On some critical and slightly super-critical sub-superlinear equations, Far East J. Appl. Math., 23 (2006), 73-90.   Google Scholar

[10]

A. Ben MabroukM. L. Ben Mohamed and K. Omrani, Finite difference approximate solutions for a mixed sub-superlinear equation, Appl. Math. Comput., 187 (2007), 1007-1016.  doi: 10.1016/j.amc.2006.09.081.  Google Scholar

[11]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equationcoupled with Maxwell equations, Rev. Math. Phys., 14 (2020), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar

[12]

R. D. BenguriaJ. Dolbeault and M. J. Esteban, Classification of the solutions of semilinear elliptic problems in a ball, J. Differential Equations, 167 (2000), 438-466.  doi: 10.1006/jdeq.2000.3792.  Google Scholar

[13]

K. Chaïb, Necessary and sufficient conditions of existence for a system involving the $p$-Laplacian $(0 < p < N)$, J. Differential Equations, 189 (2003), 513-525.  doi: 10.1016/S0022-0396(02)00094-3.  Google Scholar

[14]

S. ChakravartyM. J. AblowitzJ. R. Sauer and R. B. Jenkins, Multisoliton interactions and wavelength-division multiplexing, Opt. Lett., 20 (1995), 136-138.  doi: 10.1364/OL.20.000136.  Google Scholar

[15]

R. ChteouiA. Ben Mabrouk and H. Ounaies, Existence and properties of radial solutions of a sublinear elliptic equation, J. Partial Differ. Equ., 28 (2015), 30-38.  doi: 10.4208/jpde.v28.n1.4.  Google Scholar

[16]

A. K. Dhar and K. P. Das, Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water, Physics of Fluids A: Fluid Dynamics, 3 (1991), 3021-3026.  doi: 10.1063/1.858209.  Google Scholar

[17]

M. R. GuptaB. K. Som and B. Dasgupta, Coupled nonlinear Schrödinger equations for Langmuir and elecromagnetic waves and extension of their modulational instability domain, J. Plas, Phys., 25 (1981), 499-507.  doi: 10.1017/S0022377800026271.  Google Scholar

[18]

F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations, Phys. Rev. E, 58 (1998), 6700-6707.  doi: 10.1103/PhysRevE.58.6700.  Google Scholar

[19]

T. Kanna, M. Lakshmanan, P. Tchofo Dinda and N. Akhmediev, Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations, Phys. Rev. E, 73 (2006), 026604, 15 pp. doi: 10.1103/PhysRevE.73.026604.  Google Scholar

[20]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[21]

S. Keraani, On the blow-up phenomenon of the critical Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar

[22]

H. Liu, Ground states of linearly coupled Schrödinger systems, Electron. J. Differential Equations, (2017), Paper No. 5, 10 pp.  Google Scholar

[23]

P. Liu and S.-Y. Lou, Coupled nonlinear Schrödinger equation: Symmetries and exact solutions, Commun. Theor. Phys., 51 (2009), 27-34.  doi: 10.1088/0253-6102/51/1/06.  Google Scholar

[24]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.  Google Scholar

[25]

C. R. Menyuk, Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes, Journal of the Optical Society of America B, 5 (1988), 392-402.  doi: 10.1364/JOSAB.5.000392.  Google Scholar

[26]

F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearities, Comm. Math. Phys., 129 (1990), 223-240.  doi: 10.1007/BF02096981.  Google Scholar

[27]

L. F. MollenauerS. G. Evangelides and J. P. Gordon, Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers, J. Lightwave Technol., 9 (1991), 362-367.  doi: 10.1109/50.70013.  Google Scholar

[28]

H. Ounaies, Study of an elliptic equation with a singular potential, Indian J. Pure Appl. Math., 34 (2003), 111-131.   Google Scholar

[29]

Z. Pinar and E. Deliktas, Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities, AIP Conference Proceedings, 1815 (2017), 080019. doi: 10.1063/1.4976451.  Google Scholar

[30]

T. Saanouni, A note on coupled focusing nonlinear Schrödinger equations, Appl. Anal., 95 (2016), 2063-2080.  doi: 10.1080/00036811.2015.1086757.  Google Scholar

[31]

J. Serrin and H. Zou, Classification of positive solutions of quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 3 (1994), 1-25.  doi: 10.12775/TMNA.1994.001.  Google Scholar

[32]

M. ShalabyF. Reynaud and A. Barthelemy, Experimental observation of spatial soliton interactions with a $\pi/2$ relative phase difference, Opt. Lett., 17 (1992), 778-780.   Google Scholar

[33]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[34]

S. L. Yadava, Uniqueness of positive radial solutions of the Dirichlet problems $-\Delta u = u^p\pm u^q$ in an annulus, J. Differential Equations, 139 (1997), 194-217.  doi: 10.1006/jdeq.1997.3283.  Google Scholar

[35]

E. Yanagida, Structure of radial solutions to $\Delta u+K(|x|)|u|^{p-1}u = 0$ in $\mathbb{R}^n$, SIAM. J. Math. Anal., 27 (1996), 997-1014.  doi: 10.1137/0527053.  Google Scholar

[36]

H.-Q. ZhangX.-H. MengT. XuL.-L. Li and B. Tian, Interactions of bright solitons for the $(2+1)$-dimensional coupled nonlinear Schrödinger equations from optical fibres with symbolic computation, Phys. Scr., 75 (2007), 537-542.  doi: 10.1088/0031-8949/75/4/028.  Google Scholar

[37]

Y. Zhida, Multi-soliton solutions of coupled nonlinear Schrödinger Equations., J. Chinese Physics Letters, 4 (1987), 185-187. Google Scholar

[38]

S. Zhou and X. Cheng, Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains, Math. Comput. Simulation, 80 (2010), 2362-2373.  doi: 10.1016/j.matcom.2010.05.019.  Google Scholar

show all references

References:
[1]

J. S. AitchisonA. M. WeinerY. SilberbergM. K. OliverJ. L. JackelD. E. LeairdE. M. Vogel and P. W. E. Smith, Observation of spatial optical solitons in a nonlinear glass waveguide, Opt. Lett., 15 (1990), 471-473.  doi: 10.1364/OL.15.000471.  Google Scholar

[2]

H. AminikhahF. Pournasiri and F. Mehrdoust, A novel effective approach for systems of coupled Schrödinger equation, Pramana, 86 (2016), 19-30.  doi: 10.1007/s12043-015-0961-4.  Google Scholar

[3]

B. BalabaneJ. Dolbeault and H. Ounaeis, Nodal solutions for a sublinear elliptic equation, Nonlinear Anal., 52 (2003), 219-237.  doi: 10.1016/S0362-546X(02)00104-9.  Google Scholar

[4]

A. Ben Mabrouk and M. Ayadi, A linearized finite-difference method for the solution of some mixed concave and convex nonlinear problems, Appl. Math. Comput., 197 (2008), 1-10.  doi: 10.1016/j.amc.2007.07.051.  Google Scholar

[5]

A. Ben Mabrouk and M. Ayadi, Lyapunov type operators for numerical solutions of PDEs, Appl. Math. Comput., 204 (2008), 395-407.  doi: 10.1016/j.amc.2008.06.061.  Google Scholar

[6]

A. Ben Mabrouk and M. L. Ben Mohamed, Nodal solutions for some nonlinear elliptic equations, Appl. Math. Comput., 186 (2007), 589-597.  doi: 10.1016/j.amc.2006.08.003.  Google Scholar

[7]

A. Ben Mabrouk and M. L. Ben Mohamed, Phase plane analysis and classification of solutions of a mixed sublinear-superlinear elliptic problem, Nonlinear Anal., 70 (2009), 1-15.  doi: 10.1016/j.na.2007.11.041.  Google Scholar

[8]

A. Ben Mabrouk and M. L. Ben Mohamed, Nonradial solutions of a mixed concave-convex elliptic problem, J. Partial Differ. Equ., 24 (2011), 313-323.  doi: 10.4208/jpde.v24.n4.3.  Google Scholar

[9]

A. Ben Mabrouk and M. L. Ben Mohamed, On some critical and slightly super-critical sub-superlinear equations, Far East J. Appl. Math., 23 (2006), 73-90.   Google Scholar

[10]

A. Ben MabroukM. L. Ben Mohamed and K. Omrani, Finite difference approximate solutions for a mixed sub-superlinear equation, Appl. Math. Comput., 187 (2007), 1007-1016.  doi: 10.1016/j.amc.2006.09.081.  Google Scholar

[11]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equationcoupled with Maxwell equations, Rev. Math. Phys., 14 (2020), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar

[12]

R. D. BenguriaJ. Dolbeault and M. J. Esteban, Classification of the solutions of semilinear elliptic problems in a ball, J. Differential Equations, 167 (2000), 438-466.  doi: 10.1006/jdeq.2000.3792.  Google Scholar

[13]

K. Chaïb, Necessary and sufficient conditions of existence for a system involving the $p$-Laplacian $(0 < p < N)$, J. Differential Equations, 189 (2003), 513-525.  doi: 10.1016/S0022-0396(02)00094-3.  Google Scholar

[14]

S. ChakravartyM. J. AblowitzJ. R. Sauer and R. B. Jenkins, Multisoliton interactions and wavelength-division multiplexing, Opt. Lett., 20 (1995), 136-138.  doi: 10.1364/OL.20.000136.  Google Scholar

[15]

R. ChteouiA. Ben Mabrouk and H. Ounaies, Existence and properties of radial solutions of a sublinear elliptic equation, J. Partial Differ. Equ., 28 (2015), 30-38.  doi: 10.4208/jpde.v28.n1.4.  Google Scholar

[16]

A. K. Dhar and K. P. Das, Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water, Physics of Fluids A: Fluid Dynamics, 3 (1991), 3021-3026.  doi: 10.1063/1.858209.  Google Scholar

[17]

M. R. GuptaB. K. Som and B. Dasgupta, Coupled nonlinear Schrödinger equations for Langmuir and elecromagnetic waves and extension of their modulational instability domain, J. Plas, Phys., 25 (1981), 499-507.  doi: 10.1017/S0022377800026271.  Google Scholar

[18]

F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations, Phys. Rev. E, 58 (1998), 6700-6707.  doi: 10.1103/PhysRevE.58.6700.  Google Scholar

[19]

T. Kanna, M. Lakshmanan, P. Tchofo Dinda and N. Akhmediev, Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations, Phys. Rev. E, 73 (2006), 026604, 15 pp. doi: 10.1103/PhysRevE.73.026604.  Google Scholar

[20]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[21]

S. Keraani, On the blow-up phenomenon of the critical Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar

[22]

H. Liu, Ground states of linearly coupled Schrödinger systems, Electron. J. Differential Equations, (2017), Paper No. 5, 10 pp.  Google Scholar

[23]

P. Liu and S.-Y. Lou, Coupled nonlinear Schrödinger equation: Symmetries and exact solutions, Commun. Theor. Phys., 51 (2009), 27-34.  doi: 10.1088/0253-6102/51/1/06.  Google Scholar

[24]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.  Google Scholar

[25]

C. R. Menyuk, Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes, Journal of the Optical Society of America B, 5 (1988), 392-402.  doi: 10.1364/JOSAB.5.000392.  Google Scholar

[26]

F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearities, Comm. Math. Phys., 129 (1990), 223-240.  doi: 10.1007/BF02096981.  Google Scholar

[27]

L. F. MollenauerS. G. Evangelides and J. P. Gordon, Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers, J. Lightwave Technol., 9 (1991), 362-367.  doi: 10.1109/50.70013.  Google Scholar

[28]

H. Ounaies, Study of an elliptic equation with a singular potential, Indian J. Pure Appl. Math., 34 (2003), 111-131.   Google Scholar

[29]

Z. Pinar and E. Deliktas, Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities, AIP Conference Proceedings, 1815 (2017), 080019. doi: 10.1063/1.4976451.  Google Scholar

[30]

T. Saanouni, A note on coupled focusing nonlinear Schrödinger equations, Appl. Anal., 95 (2016), 2063-2080.  doi: 10.1080/00036811.2015.1086757.  Google Scholar

[31]

J. Serrin and H. Zou, Classification of positive solutions of quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 3 (1994), 1-25.  doi: 10.12775/TMNA.1994.001.  Google Scholar

[32]

M. ShalabyF. Reynaud and A. Barthelemy, Experimental observation of spatial soliton interactions with a $\pi/2$ relative phase difference, Opt. Lett., 17 (1992), 778-780.   Google Scholar

[33]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[34]

S. L. Yadava, Uniqueness of positive radial solutions of the Dirichlet problems $-\Delta u = u^p\pm u^q$ in an annulus, J. Differential Equations, 139 (1997), 194-217.  doi: 10.1006/jdeq.1997.3283.  Google Scholar

[35]

E. Yanagida, Structure of radial solutions to $\Delta u+K(|x|)|u|^{p-1}u = 0$ in $\mathbb{R}^n$, SIAM. J. Math. Anal., 27 (1996), 997-1014.  doi: 10.1137/0527053.  Google Scholar

[36]

H.-Q. ZhangX.-H. MengT. XuL.-L. Li and B. Tian, Interactions of bright solitons for the $(2+1)$-dimensional coupled nonlinear Schrödinger equations from optical fibres with symbolic computation, Phys. Scr., 75 (2007), 537-542.  doi: 10.1088/0031-8949/75/4/028.  Google Scholar

[37]

Y. Zhida, Multi-soliton solutions of coupled nonlinear Schrödinger Equations., J. Chinese Physics Letters, 4 (1987), 185-187. Google Scholar

[38]

S. Zhou and X. Cheng, Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains, Math. Comput. Simulation, 80 (2010), 2362-2373.  doi: 10.1016/j.matcom.2010.05.019.  Google Scholar

Figure 1.  Partition of the plane $ \mathbb{R}^2 $ according to the curves $ \Gamma_{1} $ and $ \Gamma_{2} $ for $ p = 1.5 $ and $ \omega = 2 $
Figure 2.  Partition of the plane $ \mathbb{R}^2 $ according to $ \Gamma_1 $, $ \Gamma_{2} $ and $ \Lambda $ for $ p = 1.5 $ and $ \omega = 2 $
Figure 3.  $ (u,v) $ for $ (a,b) = (0.25,2.75)\in\Omega_1 $
Figure 4.  The solution $ (u,v) $ for $ (a,b) = (1,\omega_3-0.1)\in\Omega_1 $
Figure 5.  The solutions $ u $ and $ v $ separately for $ (a,b) = (0.25,2.75)\in\Omega_1 $
Figure 6.  The phase plane portrait $ (u',u) $ for $ (a,b) = (0.25,2.75)\in\Omega_1 $
Figure 7.  The phase plane portrait $ (v',v) $ for $ (a,b) = (0.25,2.75)\in\Omega_1 $
Figure 8.  The partition of $ \Omega_2 $ according to the curve $ \Lambda $
Figure 9.  The solution $ (u,v) $ for $ (a,b) = (0.45,0.95)\in\Omega_2^1 $
Figure 10.  The solution $ (u,v) $ for $ (a,b) = (0.5,0.75)\in\Omega_2^1 $
Figure 11.  The solution $ (u,v) $ for $ (a,b) = (0.4,0.5)\in\Omega_2^1 $
Figure 12.  The solutions $ u $ an $ v $ separately for $ (a,b) = (0.45,0.5)\in\Omega_2^1 $
Figure 13.  The solution $ (u,v) $ for $ p = 2 $, $ \omega = 2 $ and $ (a,b) = (2.5,4)\in\Omega_{ext,1} $
Figure 14.  The solution $ (u,v) $ for $ p = 2 $, $ \omega = 2 $ and $ (a,b) = (1,6.5)\in\Omega_{ext,1} $
Figure 15.  The solutions $ u $ and $ v $ separately for $ p = 2 $, $ \omega = 2 $ and $ (a,b) = (0.5,5)\in\Omega_{ext,1} $
Figure 16.  The solution $ u $ and $ v $ separately for $ p = 2 $, $ \omega = 2 $ and $ (a,b) = (3.5,4)\in\Omega_{ext,1} $
Figure 17.  The solution $ (u,v) $ for $ p = 1.5 $, $ \omega = 2 $ and $ (a,b) = (0.5,3.0625)\in)A,B( $
Figure 18.  The solution $ (u,v) $ for $ p = 1.5 $, $ \omega = 2 $ and $ (a,b) = (0.9511,1.2)\in)A,B( $
Figure 19.  The solutions $ u $ and $ v $ separately for $ p = 1.5 $, $ \omega = 2 $ and $ (a,b) = (0.5,3.0625)\in)A,B( $
Figure 20.  The solutions $ u $ and $ v $ separately for $ p = 1.5 $, $ \omega = 2 $ and $ (a,b) = (0.9511,1.2)\in)A,B( $
Figure 21.  The solution $ (u,v) $ for $ p = 1.5 $, $ \omega = 2 $ and $ (a,b) = (0.1,1.2976)\in)I,B( $
Figure 22.  The solution $ (u,v) $ for $ p = 1.5 $, $ \omega = 2 $ and $ (a,b) = (0.5,1.1371)\in)I,B( $
Figure 23.  The solutions $ u $ and $ v $ separately for $ (a,b) = (0.1,1.2976)\in)I,B( $
Figure 24.  The chaotic behavior for $ p = 1.5 $, $ w = 2 $ and $ (a,b) = (0.2,0.4) $
Figure 25.  The chaotic behavior for $ p = 1.5 $, $ w = 2 $ and $ (a,b) = (2,4) $
Figure 26.  The chaotic behavior for $ p = 2.5 $, $ w = 3 $ and $ (a,b) = (0.2,0.14) $
Figure 27.  The chaotic behavior for $ p = 2.5 $, $ w = 2 $ and $ (a,b) = (2,4) $
Figure 28.  The partition of the plane according to $ \Gamma_1 $, $ \Gamma_2 $ and $ \Lambda $ for $ w = 0 $
Figure 29.  The solution $ (u,v) $ for $ p = 1.5 $, $ w = 0 $ and $ (a,b) = (0.25,0.5538)\in\Lambda $
Figure 30.  The solution $ (u,v) $ for $ p = 1.5 $, $ w = 0 $ and $ (a,b) = (0.25,0.35)\in R_2 $
Figure 31.  The solution $ (u,v) $ for $ p = 1.5 $, $ w = 0 $ and $ (a,b) = (0.5,1)\in R_1 $
Figure 32.  The solution $ (u,v) $ for $ p = 1.5 $, $ w = 0 $ and $ (a,b) = (2.5,3.5)\in R_1 $
Figure 33.  The portrait $ (u',u) $ for $ p = 1.5 $, $ w = 0 $ and $ (a,b) = (0.25,0.5538)\in\Lambda $
Figure 34.  The portrait $ (v',v) $ for $ p = 1.5 $, $ w = 0 $ and $ (a,b) = (0.25,0.35)\in R_2 $
Figure 35.  The solutions $ u $ (in blue) and $ v $ (in pink) for $ p = 1.5 $, $ w = 0 $ and $ (a,b) = (0.5,1)\in R_1 $
Figure 36.  The energy $ E(u,v)(x) $ for $ p = 1.5 $, $ (a,b) = (0.15,25)\in R_2 $, $ w = 0 $ at the top and $ w = 2 $ at the bottom
Figure 37.  Parameters' domains for $ p = 1.5 $, $ \omega = 0.5 $
Figure 38.  Parameters' domains for $ p = 3.5 $, $ \omega = 0.5 $
Figure 39.  Parameters' domains for $ p = 1.5 $, $ \omega = 1 $
Figure 40.  Parameters' domains for $ p = 3.5 $, $ \omega = 1 $
Table 1.  Some illustrative cases of problem (34) for $ p = 1.5 $
Corresponding Figure Figure 29 Figure 30 Figure 31 Figure 32
$ a $ $ 0.25 $ $ 0.25 $ $ 0.5 $ $ 2.5 $
$ b $ $ 0.5538 $ $ 0.35 $ $ 1 $ $ 3.5 $
Initial value region $ (a,b)\in\Lambda $ $ (a,b)\in R_2 $ $ (a,b)\in R_1 $ $ (a,b)\in R_1 $
Corresponding Figure Figure 29 Figure 30 Figure 31 Figure 32
$ a $ $ 0.25 $ $ 0.25 $ $ 0.5 $ $ 2.5 $
$ b $ $ 0.5538 $ $ 0.35 $ $ 1 $ $ 3.5 $
Initial value region $ (a,b)\in\Lambda $ $ (a,b)\in R_2 $ $ (a,b)\in R_1 $ $ (a,b)\in R_1 $
[1]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[2]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[3]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[4]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605

[5]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597

[6]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269

[7]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182

[8]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263

[9]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[10]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065

[11]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[12]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[13]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[14]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[15]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

[16]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027

[17]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[18]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[19]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[20]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

 Impact Factor: 0.263

Metrics

  • PDF downloads (26)
  • HTML views (92)
  • Cited by (0)

[Back to Top]