Home On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
Article Open Access

On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients

  • Bulent Kilic EMAIL logo , Mustafa Inc and Dumitru Baleanu
Published/Copyright: March 7, 2016
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 14 Issue 1

Abstract

This paper integrates dispersive optical solitons in special optical metamaterials with a time dependent coefficient. We obtained some optical solitons of the aforementioned equation. It is shown that the examined dependent coefficients are affected by the velocity of the wave. The first integral method (FIM) and ansatz method are applied to reach the optical soliton solutions of the one-dimensional nonlinear Schrödinger’s equation (NLSE) with time dependent coefficients.

Keywords: FIM; metamaterial; optical soliton
PACS: 02.30.Jr; 05.45.Yv; 02.70.Wz

1 Introduction

The dynamics of optical solitons propagating through optical fibers for trans-continental and trans-oceanic distances is governed by the nonlinear Schrödinger’s equation (NLSE). This NLSE is derived from Maxwell’s equation in electromagnetic by the aid of multiple-scale perturbation analysis. The NLSE appears, in the literature of optical solitons, with several forms of nonlinearity that depends on the context where it is studied. The best known mathematical modeling of optical systems generally is expressed by types of NLSE. The details of NLSE are given in the studies on nonlinear optics [1–8].

It is crucial to reach general solutions of these corresponding nonlinear equations. Thus, the general solutions of these equations provide much information about the character and the structure of the equations for researchers. Many effective methods have been improved to provide much information for physicians and engineers. Some of these methods are Tanh [9], G′/G-expansion [10], Jacobi elliptic function [11], functional variable [12], Hirota bilinear [13], exp-function [14], and first integral methods [15]. All of these methods are effective methods for acquiring traveling wave solutions for NPDE.

The FIM initially has been presented to the literature by solving the Burgers-KdV equation by Feng [15]. This method has been successfully implemented to NPDE and some fractional differential equations, which are a new type of equations. In recent years, many studies on this method have been made. Raslan [16] has used this method for the Fisher equation. Tascan and Bekir [17] have used this method for the Cahn-Allen equation. Abbasbandy and Shirzadi [18] have investigated the Benjamin BonaMohany equation by this method. Jafari et al. [19] and Hosseini et al. [20] have researched w.r.t. the Biswas–Milovic equation, the KP equation, and so on [21–24].

For this paper, we present the governing equation for metamaterials in Section 2. The FIM is described and applied in Section 3. In order to construct the combined soliton solutions, an ansatz approach is applied in Section 4. Lastly, we give some conclusions in the last section.

2 Governing equation

Soliton pulse propagation properties in complex materials with simultaneous negative real dielectric permittivity and magnetic permeability, also known as double negative materials, have attracted much attention in recent research. These types of materials are not found in nature, but rather need to be fabricated through material processed engineering. Therefore, these materials are called metamaterials [25]. In recent years, the model equation that describes the propagation of solitons and other waves through these metamaterial waveguides has been studied by many researchers. One of these studies is by Ebadi and co-workers: the tanh function method [26, 27]. We have used the aforementioned equation with additional terms that account for the metamaterials as

(1)iht+a(t)hxx+b(t)|h|2h=(iα(t)hx+iλ(t)(|h|2h)x+iμ(t)(|h|2)xh,+ϕ1(|h|2h)xx+ϕ2|h|2hxx+ϕ3h2hxx*

where a, b, α, λ, μ, and φj are the group velocity dispersion, Kerr law nonlinearity, coefficient of intermodal dispersion, coefficient of self-steepening, nonlinear dispersion, and real-valued constants that account for specific metamaterials which were introduced earlier and reported in [26].

3 The first integral method

The principal structures of the FIM are as follows:

Step 1. Taking into account the usual NPDE as:

(2)W(h, ht, hx,hxt, htt, hxx,...)=0

then Equation (2) transforms the ODE as

(3)L(H,H′,H″,H‴,...)=0

such that ξ = x ∓ ct and H′ = ∂H(ξ)/∂ξ.

Step 2. The following can be taken in ODE (3):

(4)h(x,t)=h(ξ).

Step 3. A new independent variable is produced by

(5)H(ξ)=h(ξ), G(ξ)=∂H(ξ)/∂ξ

which produces a new system of ODEs:

(6)∂H(ξ)/∂ξ=G(ξ),∂F(ξ)/∂ξ=P(H(ξ), G(ξ)).

Step 4. In accordance with the qualitative theory of ODEs [28], if it is possible to find the integrals for system (6), the solutions of system (6) can be obtained immediately. On account of the particular independent plane system, there does not exist any approximation that can guide how to reach its first integrals. The Division Theorem (DT) [29] presented us an idea how to reach the first integrals.

3.1 Application

Equation (1) turns into the following ODEs by using the wave variable h = H(ξ) ei[−κx+wt], where ξ = β(x - vt). The real and imaginary parts yield the following pair of relations

(7)(βν+2αβκ+αβ)Hξ+(3λβ+2βμ−2βκ(3ϕ1+ϕ2−ϕ3))H2Hξ=0,
(8)αβ2Hξξ−(w+aκ2+ακ)H+(b−λκ+κ2(ϕ1+ϕ2+ϕ3))H3_−β2(3ϕ1+ϕ2+ϕ3)H2Hξξ−6β2ϕ1HHξ2=0.

If we differentiate (7) once by ξ, we get

(9)Hξξ=−((3λβ+2βμ−2βκ(3ϕ1+ϕ2−ϕ3))(βν+2αβκ+αβ)),(H2Hξξ+2HHξ2).

Then by equating the right side of(9) to −β2(3ϕ1+ϕ2+ϕ3)H2Hξξ−6β2ϕ1HHξ2 in (8), we have the following constraint:

(10)ϕ2 = −ϕ3,βϕ1/ =  0,3λ − 6κϕ1+4κϕ3+2μ/ =  0,
(11)v=3λ−3αβ2ϕ1 − 6κϕ1 −6aβ2κϕ1+4κϕ3+2μ3β2ϕ1.

In (10) and (11), Hξξ can be replaced in (8) instead of −β2(3ϕ1+ϕ2+ϕ3)H2Hξξ−6β2ϕ1HHξ2. Then we have

(12)(1+αβ2)Hξξ−(w+aκ2+ακ)H+(b −λκ+κ2(ϕ1+ϕ2+ϕ3))H3=0.

Then with another transformation Hξ = G, we have

(13)Hξ = G,Gξ =(w+aκ2+ακ)(1+αβ2)H−(b−λκ+κ2(ϕ1+ϕ2+ϕ3))(1 + αβ2)H3.

In accordance with the FIM, it is supposed that H (ξ) and G (ξ) are non-trivial solutions of Equation (13) and F(H,G)=∑i=0rai(H)Gi is an irreducible function in the domain C[H, G ] such that

(14)F(H(ξ), G(ξ)) = ∑​i=0rai(H)Gi = 0,

where ai(H), (i = 0, 1, 2,... , r) are polynomials of H and ar(H)/ = 0. Equation (12) is the first integral for system (13), owing to the DT, there exists g(H) + f(H)G in C[H, G] as:

(15)dF/dξ = dFdHdHdξ+dFdGdGdξ=[g(H)+f(H)G]∑i=0rai(H)Gi.

Here, we only consider r = 1 in Equation (15).

If we equate the coefficients of Gi(i = 0, 1, 2,... , r) of Equation (15) for r = 1, we have

(16)a˙1(H) = a1(H)g(H)
(17)a˙0(H) = a1(H)g(H)+h(H)a0(H)
(18)a0(H)g(H) =a1(H)[(w+aκ2+ακ)(1 + αβ2)H−b − λκ+κ2(ϕ1+ϕ2+ϕ3))(1+αβ2)H3]

Since ai(H)(i = 0, 1) is a polynomial of H, a1(H) is a constant and h(H) = 0 from (16). For convenience, let a1(H) = 1, and equalizing the degrees of g(H) and a0(H) we conclude the degree of g(H) is equal to one. Then, we assume that g(H) = A1 + 2A2H, and we obtain the following from Equations (17) and (18):

(19)a0(H) = A2H2+A1H + A0.

Replacing a0(H), a1(H) and g(H) in Equation (18), to separate the common factors of the same terms, then equating the coefficients of H i to zero, we have the following case:

(20)A1 = 0,A2 = ± (λκ − b−κ2(ϕ1+ϕ2+ϕ3))2(1+αβ2),A0 = a (w+ακ2+ακ)2(1+αβ2)(λκ−b−κ2(ϕ1+ϕ2+ϕ3)).

Putting (20) into (14), we have

(21)Hξ = ± (w+aκ2+ακ)2(1+αβ2)(λκ−b−κ2(ϕ1+ϕ2+ϕ3)) ± (λκ − b−κ2(ϕ1+ϕ2+ϕ3))2(1+αβ2)H2(ξ).

If we solve the Equations (21), we have the following dark soliton solution

(22)H = −w+ aκ2 + ακλκ−b−κ2 (ϕ1+ϕ2+ϕ3)tanh [w+ aκ2 + ακ2(1 + αβ2)],

and the original solution of Equation (2) is

(23)h(x, t)=(−w+aκ2+ακλκ−b−κ2(ϕ1+ϕ2+ϕ3)tanh[w+aκ2+ακ2(1+αβ2)β(x−vt)])ei[−κx+wt].

4. The Ansatz approach

We use an ansatz approach to seek other types of soliton solutions of Equation (1).

First, Equation (12) will be integrated to reach the combined bright-dark [30] soliton solution of Equation (1). So we will seek a solution of the following form

(24)H(ξ) = θ0 sech [ξ] - iθ1 tanh [ξ],

where θ0 and θ1 are amplitudesof the bright and dark solitons, respectively.

By substituting (22) into (12) and setting the coeffi-cients of each term of sechi [ξ] tanhj [ξ] (i, j = 0, 1, 2) to zero we get the following relations:

(25)θ0 = θ1= ±1+ αβ22(b−λκ +κ2(ϕ1+ϕ2+ϕ3)),w=−12 (1 + αβ2+ 2ακ + iκ2).

From (25) the combined bright-dark soliton solution of Equation (1) is obtained:

(26)h(x, t)= ±1+ αβ22(b−λκ +κ2(ϕ1+ϕ2+ϕ3))(sech [β(x - vt)] - i tanh [β(x - vt)])ei[−κx−12(1 + αβ2+ 2ακ + iκ2)t].

Second, Equation (12) will be integrated to reach the combined-dark soliton solution of Equation (1). So we will seek a solution of the following form

(27)H(ξ)  θ0 tanh [ξ] −iθ1 sech [ξ],

where θ0 and θ1 (θ0 > 0, θ1 > 0) are the amplitudes of the dark and bright solitons, respectively.

By substituting (27) into (12) and setting the coeffi-cients of each term of sechi [ξ] tanhj [ξ] (i, j = 0, 1, 2) to zero we get the following relations:

(28)θ0 = θ1 = ±−1 −αβ22(b − λκ +κ2(ϕ1+ϕ2+ϕ3)),w = −12(1+αβ2 + 2ακ + aκ2).

From (28) it is the combined-dark soliton solution of Equation (1) is obtained:

(29)h(x,t)  = ±−1 −αβ22(b − λκ +κ2(ϕ1+ϕ2+ϕ3))(tanh[β(x−v(t)] − i sech [β(x − vt)])ei[−κx−12(1+αβ2+2ακ+aκ2)t].

Third, Equation (12) will be integrated to reach the combined-bright soliton solution of Equation (1). So we will seek a solution of the following form

(30)H(ξ) = θ0 tanh[ξ]+iθ1sech[ξ],

where θ0 and θ1 (θ0 > 0, θ1 > 0) are the amplitudes of the dark and bright solitons, respectively.

By substituting (30) into (12) and setting the coeffi-cients of each term of sechi [ξ] tanhj [ξ] (i, j = 0, 1, 2) to zero we get the following relations as (28) and the combinedbright soliton solution of Equation (1) is obtained:

(31)h(x,t)  = ±−1 −αβ22(b − λκ +κ2(ϕ1+ϕ2+ϕ3))(tanh[β(x−v(t)] − i sech [β(x − vt)])ei[−κx−12(1+αβ2+2ακ+aκ2)t].

5. Conclusion

We used the FIM and antsatz approaches for acquiring several new exact solutions of the one-dimensional NLSE with time dependent coefficients. We have acquired different types of exact solutions which are dark, combined-bright, and combined-dark optical solitons. These obtained solutions are new according to our research of the literature. It has been shown that the velocity function w(t) is related to the group velocity term a(t) in (28). Consequently, the FIM and ansatz approaches are crucial ones to construct different types of the exact solutions of the NPDE and systems.

References

1 R. Kohl, D. Milovic, E. Zerrad, A. Biswas, Opt. Laser Technol. 40, 647 (2008)10.1016/j.optlastec.2007.10.002Search in Google Scholar

2 H. Leblond, D. Mihalache, Phys. Rep. 523, 61 (2013)10.1016/j.physrep.2012.10.006Search in Google Scholar

3 H. Leblond, H. Triki, D. Mihalache, Rom. Rep. Phys. 65, 925 (2013)Search in Google Scholar

4 D. Mihalache, Rom. J. Phys. 59, 295 (2014)10.1016/j.denabs.2014.08.022Search in Google Scholar

5 D. Mihalache, D. Mazilu, L.C. Crasovan, B.A. Malomed, F. Lederer, Phys. Rev. E 61, 7142 (2000)10.1103/PhysRevE.61.7142Search in Google Scholar

6 P. Masemola, A.H. Kara, A. Biswas, Opt. Laser Technol.. 45, 402 (2013)10.1016/j.optlastec.2012.06.017Search in Google Scholar

7 A. Biswas et al., Opt. Laser Technol. 44, 263 (2012)10.1016/j.optlastec.2011.07.001Search in Google Scholar

8 L. Girgis L, K.R. Khan, D. Milovic, S.H. Crutcher, S. Konar, A. Biswas, Opt. Laser Technol. 44, 1219 (2012)10.1016/j.optlastec.2012.01.007Search in Google Scholar

9 E.J. Parkes, B.R. Duffy, Comp. Phys. Commun. 98, 288 (1996)10.1016/0010-4655(96)00104-XSearch in Google Scholar

10 M. Wang, X. Li, J. Zhang, Phys. Lett. A 372, 417 (2008)10.1016/j.physleta.2007.07.051Search in Google Scholar

11 Z. Fu, S. Liu, S. Liu, Q. Zhao, Phys. Lett. A 290, 72 (2001)10.1016/S0375-9601(01)00644-2Search in Google Scholar

12 A. Zerarka, S. Ouamane, A. Attaf, Appl. Math. Comput. 217, 2897 (2010)10.1016/j.amc.2010.08.070Search in Google Scholar

13 X.B. Hu, W.X. Ma, Phys. Lett. A 293, 161 (2002)10.1016/S0375-9601(01)00850-7Search in Google Scholar

14 X.H. Wu, J.H. He, Comput.Math. Appl. 54, 966 (2007)10.1016/j.camwa.2006.12.041Search in Google Scholar

15 Z.S. Feng, J. Phys. A 35, 343 (2002)10.1088/0305-4470/35/2/312Search in Google Scholar

16 K.R. Raslan, Nonlinear Dyn. 53, 281 (2008)10.1007/s11071-007-9262-xSearch in Google Scholar

17 F. Tascan, A. Bekir, Appl. Math. Comput. 207, 279 (2009)10.1016/j.amc.2008.10.031Search in Google Scholar

18 S. Abbasbandy, A. Shirzadi, Commun. Nonlin. Sci. 15, 1759 (2010)10.1016/j.cnsns.2009.08.003Search in Google Scholar

19 H. Jafari, A. Sooraki, C.M. Khalique, Optik - Int. J. Light Electr. Optics 124, 3929 (2013)10.1016/j.ijleo.2012.11.039Search in Google Scholar

20 K. Hosseini, R. Ansari, P. Gholamin, J. Math. Anal. Appl. 387, 807 (2012)10.1016/j.jmaa.2011.09.044Search in Google Scholar

21 Y. He, S. Li, Y. Long, Diff. Equ. Dyn. Syst. 21, 199 (2013)10.1007/s12591-012-0145-3Search in Google Scholar

22 I. Aslan, Pramana 76, 533 (2011)10.1007/s12043-011-0062-ySearch in Google Scholar

23 A. Bekir, O. Ünsal, Pramana 79, 3 (2012)10.1007/s12043-012-0282-9Search in Google Scholar

24 A. Bekir, Ö. Güner, Ö. Unsal, J. Comp. Nonlin. Dyn. 10, 021020 (2015)10.1115/1.4028065Search in Google Scholar

25 V.M. Shalaev, Nature Photonics. 1, 41 (2007)10.1038/nphoton.2006.49Search in Google Scholar

26 G. Ebadi et al., Optelectron. Adv. Mat. 8, 828 (2014)Search in Google Scholar

27 A. Biswas, K.R. Khan, M.F. Mahmood, M. Belic, Optik. 125, 3299 (2014)10.1016/j.ijleo.2013.12.061Search in Google Scholar

28 T.R. Ding, C.Z. Li, Ordinary Differential Equations (Peking University Press, Peking, 1996)Search in Google Scholar

29 T.S. Zhang, Phys. Lett. A 371, 65 (2007)10.1016/j.physleta.2007.05.091Search in Google Scholar

30 Q. Zhou et al., Optelectron. Adv. Mat. 8, 837 (2014)10.1002/adma.19960081016Search in Google Scholar

Received: 2015-10-26
Accepted: 2015-12-6
Published Online: 2016-3-7
Published in Print: 2016-1-1

2016 B. Kilic et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

  1. Regular articles
  2. Speeding of α Decay in Strong Laser Fields
  3. Regular articles
  4. Multi-soliton rational solutions for some nonlinear evolution equations
  5. Regular articles
  6. Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution
  7. Regular articles
  8. Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients
  9. Regular articles
  10. Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system
  11. Regular articles
  12. Modeling the potential energy field caused by mass density distribution with Eton approach
  13. Regular articles
  14. Climate Solutions based on advanced scientific discoveries of Allatra physics
  15. Regular articles
  16. Investigation of TLD-700 energy response to low energy x-ray encountered in diagnostic radiology
  17. Regular articles
  18. Synthesis of Pt nanowires with the participation of physical vapour deposition
  19. Regular articles
  20. Quantum discord and entanglement in grover search algorithm
  21. Regular articles
  22. On order statistics from nonidentical discrete random variables
  23. Regular articles
  24. Charmed hadron photoproduction at COMPASS
  25. Regular articles
  26. Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall
  27. Regular articles
  28. Flap motion of helicopter rotors with novel, dynamic stall model
  29. Regular articles
  30. Impact of severe cracked germanium (111) substrate on aluminum indium gallium phosphate light-emitting-diode’s electro-optical performance
  31. Regular articles
  32. Slow-fast effect and generation mechanism of brusselator based on coordinate transformation
  33. Regular articles
  34. Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
  35. Regular articles
  36. Recent Progress in Search for Dark Sector Signatures
  37. Regular articles
  38. Recent progress in organic spintronics
  39. Regular articles
  40. On the Construction of a Surface Family with Common Geodesic in Galilean Space G3
  41. Regular articles
  42. Self-healing phenomena of graphene: potential and applications
  43. Regular articles
  44. Viscous flow and heat transfer over an unsteady stretching surface
  45. Regular articles
  46. Spacetime Exterior to a Star: Against Asymptotic Flatness
  47. Regular articles
  48. Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation
  49. Regular articles
  50. Corrosion and mechanical properties of AM50 magnesium alloy after modified by different amounts of rare earth element Gadolinium
  51. Regular articles
  52. Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations
  53. Regular articles
  54. Energy and Wave function Analysis on Harmonic Oscillator Under Simultaneous Non-Hermitian Transformations of Co-ordinate and Momentum: Iso-spectral case
  55. Regular articles
  56. Unification of all hyperbolic tangent function methods
  57. Regular articles
  58. Analytical solution for the correlator with Gribov propagators
  59. Regular articles
  60. A New Algorithm for the Approximation of the Schrödinger Equation
  61. Regular articles
  62. Analytical solutions for the fractional diffusion-advection equation describing super-diffusion
  63. Regular articles
  64. On the fractional differential equations with not instantaneous impulses
  65. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  66. Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method
  67. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  68. Numerical solution of two dimensional time fractional-order biological population model
  69. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  70. Rotational surfaces in isotropic spaces satisfying weingarten conditions
  71. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  72. Anti-synchronization of fractional order chaotic and hyperchaotic systems with fully unknown parameters using modified adaptive control
  73. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  74. Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime
  75. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  76. Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group
  77. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  78. Logical entropy of quantum dynamical systems
  79. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  80. An efficient algorithm for solving fractional differential equations with boundary conditions
  81. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  82. A numerical method for solving systems of higher order linear functional differential equations
  83. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  84. Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
  85. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  86. On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
  87. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  88. On soliton solutions of the Wu-Zhang system
  89. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  90. Comparison between the (G’/G) - expansion method and the modified extended tanh method
  91. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  92. On the union of graded prime ideals
  93. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  94. Oscillation criteria for nonlinear fractional differential equation with damping term
  95. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  96. A new method for computing the reliability of consecutive k-out-of-n:F systems
  97. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  98. A time-delay equation: well-posedness to optimal control
  99. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  100. Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
  101. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  102. Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
  103. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  104. The calculation of the optical gap energy of ZnXO (X = Bi, Sn and Fe)
  105. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  106. Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal
  107. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  108. A certain sequence of functions involving the Aleph function
  109. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  110. On negacyclic codes over the ring ℤp + uℤp + . . . + uk + 1 ℤp
  111. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  112. Solitary and compacton solutions of fractional KdV-like equations
  113. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  114. Regarding on the exact solutions for the nonlinear fractional differential equations
  115. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  116. Non-local Integrals and Derivatives on Fractal Sets with Applications
  117. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  118. On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method
  119. Special issue on Information Technology and Computational Physics
  120. On uninorms and nullnorms on direct product of bounded lattices
  121. Special issue on Information Technology and Computational Physics
  122. Phase-space description of the coherent state dynamics in a small one-dimensional system
  123. Special issue on Information Technology and Computational Physics
  124. Automated Program Design – an Example Solving a Weather Forecasting Problem
  125. Special issue on Information Technology and Computational Physics
  126. Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation
  127. Special issue on Information Technology and Computational Physics
  128. Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions
  129. Special issue on Information Technology and Computational Physics
  130. Application of Genetic Algorithm and Particle Swarm Optimization techniques for improved image steganography systems
  131. Special issue on Information Technology and Computational Physics
  132. Intelligent Chatter Bot for Regulation Search
  133. Special issue on Information Technology and Computational Physics
  134. Modeling and optimization of Quality of Service routing in Mobile Ad hoc Networks
  135. Special issue on Information Technology and Computational Physics
  136. Resource management for server virtualization under the limitations of recovery time objective
  137. Special issue on Information Technology and Computational Physics
  138. MODY – calculation of ordered structures by symmetry-adapted functions
  139. Special issue on Information Technology and Computational Physics
  140. Survey of Object-Based Data Reduction Techniques in Observational Astronomy
  141. Special issue on Information Technology and Computational Physics
  142. Optimization of the prediction of second refined wavelet coefficients in electron structure calculations
  143. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  144. Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study
  145. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  146. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
  147. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  148. Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
  149. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  150. Numerical heat transfer analysis of transcritical hydrocarbon fuel flow in a tube partially filled with porous media
  151. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  152. Experimental Investigation on Oil Enhancement Mechanism of Hot Water Injection in tight reservoirs
  153. Special Issue on Research Frontier on Molecular Reaction Dynamics
  154. Role of intramolecular hydrogen bonding in the excited-state intramolecular double proton transfer (ESIDPT) of calix[4]arene: A TDDFT study
  155. Special Issue on Research Frontier on Molecular Reaction Dynamics
  156. Hydrogen-bonding study of photoexcited 4-nitro-1,8-naphthalimide in hydrogen-donating solvents
  157. Special Issue on Research Frontier on Molecular Reaction Dynamics
  158. The Interaction between Graphene and Oxygen Atom
  159. Special Issue on Research Frontier on Molecular Reaction Dynamics
  160. Kinetics of the austenitization in the Fe-Mo-C ternary alloys during continuous heating
  161. Special Issue: Functional Advanced and Nanomaterials
  162. Colloidal synthesis of Culn0.75Ga0.25Se2 nanoparticles and their photovoltaic performance
  163. Special Issue: Functional Advanced and Nanomaterials
  164. Positioning and aligning CNTs by external magnetic field to assist localised epoxy cure
  165. Special Issue: Functional Advanced and Nanomaterials
  166. Quasi-planar elemental clusters in pair interactions approximation
  167. Special Issue: Functional Advanced and Nanomaterials
  168. Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate
https://doi.org/10.1515/phys-2016-0003
Keywords for this article
FIM; metamaterial; optical soliton
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Regular articles
  2. Speeding of α Decay in Strong Laser Fields
  3. Regular articles
  4. Multi-soliton rational solutions for some nonlinear evolution equations
  5. Regular articles
  6. Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution
  7. Regular articles
  8. Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients
  9. Regular articles
  10. Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system
  11. Regular articles
  12. Modeling the potential energy field caused by mass density distribution with Eton approach
  13. Regular articles
  14. Climate Solutions based on advanced scientific discoveries of Allatra physics
  15. Regular articles
  16. Investigation of TLD-700 energy response to low energy x-ray encountered in diagnostic radiology
  17. Regular articles
  18. Synthesis of Pt nanowires with the participation of physical vapour deposition
  19. Regular articles
  20. Quantum discord and entanglement in grover search algorithm
  21. Regular articles
  22. On order statistics from nonidentical discrete random variables
  23. Regular articles
  24. Charmed hadron photoproduction at COMPASS
  25. Regular articles
  26. Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall
  27. Regular articles
  28. Flap motion of helicopter rotors with novel, dynamic stall model
  29. Regular articles
  30. Impact of severe cracked germanium (111) substrate on aluminum indium gallium phosphate light-emitting-diode’s electro-optical performance
  31. Regular articles
  32. Slow-fast effect and generation mechanism of brusselator based on coordinate transformation
  33. Regular articles
  34. Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
  35. Regular articles
  36. Recent Progress in Search for Dark Sector Signatures
  37. Regular articles
  38. Recent progress in organic spintronics
  39. Regular articles
  40. On the Construction of a Surface Family with Common Geodesic in Galilean Space G3
  41. Regular articles
  42. Self-healing phenomena of graphene: potential and applications
  43. Regular articles
  44. Viscous flow and heat transfer over an unsteady stretching surface
  45. Regular articles
  46. Spacetime Exterior to a Star: Against Asymptotic Flatness
  47. Regular articles
  48. Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation
  49. Regular articles
  50. Corrosion and mechanical properties of AM50 magnesium alloy after modified by different amounts of rare earth element Gadolinium
  51. Regular articles
  52. Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations
  53. Regular articles
  54. Energy and Wave function Analysis on Harmonic Oscillator Under Simultaneous Non-Hermitian Transformations of Co-ordinate and Momentum: Iso-spectral case
  55. Regular articles
  56. Unification of all hyperbolic tangent function methods
  57. Regular articles
  58. Analytical solution for the correlator with Gribov propagators
  59. Regular articles
  60. A New Algorithm for the Approximation of the Schrödinger Equation
  61. Regular articles
  62. Analytical solutions for the fractional diffusion-advection equation describing super-diffusion
  63. Regular articles
  64. On the fractional differential equations with not instantaneous impulses
  65. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  66. Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method
  67. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  68. Numerical solution of two dimensional time fractional-order biological population model
  69. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  70. Rotational surfaces in isotropic spaces satisfying weingarten conditions
  71. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  72. Anti-synchronization of fractional order chaotic and hyperchaotic systems with fully unknown parameters using modified adaptive control
  73. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  74. Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime
  75. Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
  76. Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group
  77. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  78. Logical entropy of quantum dynamical systems
  79. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  80. An efficient algorithm for solving fractional differential equations with boundary conditions
  81. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  82. A numerical method for solving systems of higher order linear functional differential equations
  83. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  84. Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
  85. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  86. On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
  87. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  88. On soliton solutions of the Wu-Zhang system
  89. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  90. Comparison between the (G’/G) - expansion method and the modified extended tanh method
  91. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  92. On the union of graded prime ideals
  93. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  94. Oscillation criteria for nonlinear fractional differential equation with damping term
  95. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  96. A new method for computing the reliability of consecutive k-out-of-n:F systems
  97. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  98. A time-delay equation: well-posedness to optimal control
  99. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  100. Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
  101. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  102. Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
  103. Topical Issue: Recent Developments in Applied and Engineering Mathematics
  104. The calculation of the optical gap energy of ZnXO (X = Bi, Sn and Fe)
  105. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  106. Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal
  107. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  108. A certain sequence of functions involving the Aleph function
  109. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  110. On negacyclic codes over the ring ℤp + uℤp + . . . + uk + 1 ℤp
  111. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  112. Solitary and compacton solutions of fractional KdV-like equations
  113. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  114. Regarding on the exact solutions for the nonlinear fractional differential equations
  115. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  116. Non-local Integrals and Derivatives on Fractal Sets with Applications
  117. Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
  118. On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method
  119. Special issue on Information Technology and Computational Physics
  120. On uninorms and nullnorms on direct product of bounded lattices
  121. Special issue on Information Technology and Computational Physics
  122. Phase-space description of the coherent state dynamics in a small one-dimensional system
  123. Special issue on Information Technology and Computational Physics
  124. Automated Program Design – an Example Solving a Weather Forecasting Problem
  125. Special issue on Information Technology and Computational Physics
  126. Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation
  127. Special issue on Information Technology and Computational Physics
  128. Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions
  129. Special issue on Information Technology and Computational Physics
  130. Application of Genetic Algorithm and Particle Swarm Optimization techniques for improved image steganography systems
  131. Special issue on Information Technology and Computational Physics
  132. Intelligent Chatter Bot for Regulation Search
  133. Special issue on Information Technology and Computational Physics
  134. Modeling and optimization of Quality of Service routing in Mobile Ad hoc Networks
  135. Special issue on Information Technology and Computational Physics
  136. Resource management for server virtualization under the limitations of recovery time objective
  137. Special issue on Information Technology and Computational Physics
  138. MODY – calculation of ordered structures by symmetry-adapted functions
  139. Special issue on Information Technology and Computational Physics
  140. Survey of Object-Based Data Reduction Techniques in Observational Astronomy
  141. Special issue on Information Technology and Computational Physics
  142. Optimization of the prediction of second refined wavelet coefficients in electron structure calculations
  143. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  144. Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study
  145. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  146. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
  147. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  148. Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
  149. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  150. Numerical heat transfer analysis of transcritical hydrocarbon fuel flow in a tube partially filled with porous media
  151. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  152. Experimental Investigation on Oil Enhancement Mechanism of Hot Water Injection in tight reservoirs
  153. Special Issue on Research Frontier on Molecular Reaction Dynamics
  154. Role of intramolecular hydrogen bonding in the excited-state intramolecular double proton transfer (ESIDPT) of calix[4]arene: A TDDFT study
  155. Special Issue on Research Frontier on Molecular Reaction Dynamics
  156. Hydrogen-bonding study of photoexcited 4-nitro-1,8-naphthalimide in hydrogen-donating solvents
  157. Special Issue on Research Frontier on Molecular Reaction Dynamics
  158. The Interaction between Graphene and Oxygen Atom
  159. Special Issue on Research Frontier on Molecular Reaction Dynamics
  160. Kinetics of the austenitization in the Fe-Mo-C ternary alloys during continuous heating
  161. Special Issue: Functional Advanced and Nanomaterials
  162. Colloidal synthesis of Culn0.75Ga0.25Se2 nanoparticles and their photovoltaic performance
  163. Special Issue: Functional Advanced and Nanomaterials
  164. Positioning and aligning CNTs by external magnetic field to assist localised epoxy cure
  165. Special Issue: Functional Advanced and Nanomaterials
  166. Quasi-planar elemental clusters in pair interactions approximation
  167. Special Issue: Functional Advanced and Nanomaterials
  168. Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2016-0003/html
Scroll to top button