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A C0 interior penalty method for the Cahn-Hilliard equation
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
H. Liu, Ground states of linearly coupled Schrödinger systems, Electron. J. Differential Equations, (2017), Paper No. 5, 10 pp. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
H. Ounaies,
Study of an elliptic equation with a singular potential, Indian J. Pure Appl. Math., 34 (2003), 111-131.
|
[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. |
[30] |
T. Saanouni,
A note on coupled focusing nonlinear Schrödinger equations, Appl. Anal., 95 (2016), 2063-2080.
doi: 10.1080/00036811.2015.1086757. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
H. Liu, Ground states of linearly coupled Schrödinger systems, Electron. J. Differential Equations, (2017), Paper No. 5, 10 pp. |
[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. |
[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. |
[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. |
[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. |
[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. |
[28] |
H. Ounaies,
Study of an elliptic equation with a singular potential, Indian J. Pure Appl. Math., 34 (2003), 111-131.
|
[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. |
[30] |
T. Saanouni,
A note on coupled focusing nonlinear Schrödinger equations, Appl. Anal., 95 (2016), 2063-2080.
doi: 10.1080/00036811.2015.1086757. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |

















Corresponding Figure | Figure 29 | Figure 30 | Figure 31 | Figure 32 |
Initial value region |
Corresponding Figure | Figure 29 | Figure 30 | Figure 31 | Figure 32 |
Initial value region |
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