# American Institute of Mathematical Sciences

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. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. 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. Aminikhah, F. 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. Balabane, J. 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 Mabrouk, M. 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. Benguria, J. 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. Chakravarty, M. J. Ablowitz, J. 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. Chteoui, A. 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. Gupta, B. 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. Mollenauer, S. 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. Shalaby, F. 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. Zhang, X.-H. Meng, T. Xu, L.-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. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. 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. Aminikhah, F. 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. Balabane, J. 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 Mabrouk, M. 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. Benguria, J. 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. Chakravarty, M. J. Ablowitz, J. 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. Chteoui, A. 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. Gupta, B. 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. Mollenauer, S. 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. Shalaby, F. 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. Zhang, X.-H. Meng, T. Xu, L.-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
Partition of the plane $\mathbb{R}^2$ according to the curves $\Gamma_{1}$ and $\Gamma_{2}$ for $p = 1.5$ and $\omega = 2$
Partition of the plane $\mathbb{R}^2$ according to $\Gamma_1$, $\Gamma_{2}$ and $\Lambda$ for $p = 1.5$ and $\omega = 2$
$(u,v)$ for $(a,b) = (0.25,2.75)\in\Omega_1$
The solution $(u,v)$ for $(a,b) = (1,\omega_3-0.1)\in\Omega_1$
The solutions $u$ and $v$ separately for $(a,b) = (0.25,2.75)\in\Omega_1$
The phase plane portrait $(u',u)$ for $(a,b) = (0.25,2.75)\in\Omega_1$
The phase plane portrait $(v',v)$ for $(a,b) = (0.25,2.75)\in\Omega_1$
The partition of $\Omega_2$ according to the curve $\Lambda$
The solution $(u,v)$ for $(a,b) = (0.45,0.95)\in\Omega_2^1$
The solution $(u,v)$ for $(a,b) = (0.5,0.75)\in\Omega_2^1$
The solution $(u,v)$ for $(a,b) = (0.4,0.5)\in\Omega_2^1$
The solutions $u$ an $v$ separately for $(a,b) = (0.45,0.5)\in\Omega_2^1$
The solution $(u,v)$ for $p = 2$, $\omega = 2$ and $(a,b) = (2.5,4)\in\Omega_{ext,1}$
The solution $(u,v)$ for $p = 2$, $\omega = 2$ and $(a,b) = (1,6.5)\in\Omega_{ext,1}$
The solutions $u$ and $v$ separately for $p = 2$, $\omega = 2$ and $(a,b) = (0.5,5)\in\Omega_{ext,1}$
The solution $u$ and $v$ separately for $p = 2$, $\omega = 2$ and $(a,b) = (3.5,4)\in\Omega_{ext,1}$
The solution $(u,v)$ for $p = 1.5$, $\omega = 2$ and $(a,b) = (0.5,3.0625)\in)A,B($
The solution $(u,v)$ for $p = 1.5$, $\omega = 2$ and $(a,b) = (0.9511,1.2)\in)A,B($
The solutions $u$ and $v$ separately for $p = 1.5$, $\omega = 2$ and $(a,b) = (0.5,3.0625)\in)A,B($
The solutions $u$ and $v$ separately for $p = 1.5$, $\omega = 2$ and $(a,b) = (0.9511,1.2)\in)A,B($
The solution $(u,v)$ for $p = 1.5$, $\omega = 2$ and $(a,b) = (0.1,1.2976)\in)I,B($
The solution $(u,v)$ for $p = 1.5$, $\omega = 2$ and $(a,b) = (0.5,1.1371)\in)I,B($
The solutions $u$ and $v$ separately for $(a,b) = (0.1,1.2976)\in)I,B($
The chaotic behavior for $p = 1.5$, $w = 2$ and $(a,b) = (0.2,0.4)$
The chaotic behavior for $p = 1.5$, $w = 2$ and $(a,b) = (2,4)$
The chaotic behavior for $p = 2.5$, $w = 3$ and $(a,b) = (0.2,0.14)$
The chaotic behavior for $p = 2.5$, $w = 2$ and $(a,b) = (2,4)$
The partition of the plane according to $\Gamma_1$, $\Gamma_2$ and $\Lambda$ for $w = 0$
The solution $(u,v)$ for $p = 1.5$, $w = 0$ and $(a,b) = (0.25,0.5538)\in\Lambda$
The solution $(u,v)$ for $p = 1.5$, $w = 0$ and $(a,b) = (0.25,0.35)\in R_2$
The solution $(u,v)$ for $p = 1.5$, $w = 0$ and $(a,b) = (0.5,1)\in R_1$
The solution $(u,v)$ for $p = 1.5$, $w = 0$ and $(a,b) = (2.5,3.5)\in R_1$
The portrait $(u',u)$ for $p = 1.5$, $w = 0$ and $(a,b) = (0.25,0.5538)\in\Lambda$
The portrait $(v',v)$ for $p = 1.5$, $w = 0$ and $(a,b) = (0.25,0.35)\in R_2$
The solutions $u$ (in blue) and $v$ (in pink) for $p = 1.5$, $w = 0$ and $(a,b) = (0.5,1)\in R_1$
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
Parameters' domains for $p = 1.5$, $\omega = 0.5$
Parameters' domains for $p = 3.5$, $\omega = 0.5$
Parameters' domains for $p = 1.5$, $\omega = 1$
Parameters' domains for $p = 3.5$, $\omega = 1$
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 $(a,b)\in\Lambda$ $(a,b)\in R_2$ $(a,b)\in R_1$ $(a,b)\in R_1$
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