Optical soliton solutions for the Gerdjikov-Ivanov model via tan(ϕ/2)-expansion method

Abstract In this paper, by introducing new approach, the improved tan( ϕ ( ξ )/2)-expansion method (ITEM) is further extended into Gerdjikov–Ivanov (GI) model. As a result, the hyperbolic function solution, the trigonometric function solution, the exponential solution and the rational solution with free parameters are obtained. When the parameters are taken as special values the solitary wave solutions and the periodic wave solutions are also derived from the traveling wave solutions. Moreover, it is observed that the suggested technique is compatible of such problems. We obtained the further solutions comparing with other methods. The exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. We will investigate the GI envelope solitons with the framework of Madelung fluid description. The results of applying this procedure to the studied cases show the high efficiency of the new technique. It is shown that this method is a powerful mathematical tool for solving problems in fluids mechanics.

[1]  George Csordas,et al.  Basic partial differential equations , 1992 .

[2]  J. Manafian,et al.  Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics , 2016 .

[3]  J. Manafian,et al.  Solitary wave and periodic wave solutions for Burgers, Fisher, Huxley and combined forms of these equations by the (G′/G)-expansion method , 2015 .

[4]  Alexey V. Porubov,et al.  Some general periodic solutions to coupled nonlinear Schrödinger equations , 1999 .

[5]  H. M. Baskonus,et al.  Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics , 2016 .

[6]  J. Manafian,et al.  Abundant soliton solutions for the coupled Schrödinger-Boussinesq system via an analytical method , 2016 .

[7]  X. Geng,et al.  Explicit solutions of the 2 + 1-dimensional breaking soliton equation , 2004 .

[8]  S. Nicola,et al.  Solitary Waves in a Madelung Fluid Description of Derivative NLS Equations , 2008 .

[9]  Zuntao Fu,et al.  JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND PERIODIC WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS , 2001 .

[10]  Mohammad Mehdi Rashidi,et al.  A study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method , 2013 .

[11]  Chauncey D. Leake,et al.  British Association for the Advancement of Science , 1953, Science.

[12]  M. Ivanov,et al.  THE QUADRATIC BUNDLE OF GENERAL FORM AND THE NONLINEAR EVOLUTION EQUATIONS: HIERARCHIES OF HAMILTONIAN STRUCTURES , 1982 .

[13]  M. Dehghan,et al.  Application of semi‐analytic methods for the Fitzhugh–Nagumo equation, which models the transmission of nerve impulses , 2010 .

[14]  M. Dehghan,et al.  Application of the Exp‐function method for solving a partial differential equation arising in biology and population genetics , 2011 .

[15]  J. Manafian.  Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan(Φ(ξ)/2)-expansion method , 2016 .

[16]  Engui Fan,et al.  Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation , 2000 .

[17]  J. Manafian,et al.  Application of tan(ϕ/2)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity , 2016 .

[18]  Engui Fan,et al.  Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation , 2000 .

[19]  Anjan Biswas,et al.  1-Soliton solution of the Klein-Gordon-Schrodinger's equation with power law nonlinearity , 2010, Appl. Math. Comput..

[20]  P. Shukla,et al.  Solitons in the Madelung's Fluid , 2001 .

[21]  Chaudry Masood Khalique,et al.  Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model , 2015 .

[22]  J. Manafian,et al.  Study of the Analytical Treatment of the (2+1)-Dimensional Zoomeron, the Duffing and the SRLW Equations via a New Analytical Approach , 2016 .

[23]  J. Manafian,et al.  Abundant soliton solutions for the Kundu–Eckhaus equation via tan(ϕ(ξ))-expansion method , 2016 .

[24]  J. David Logan,et al.  An Introduction to Nonlinear Partial Differential Equations , 1994 .

[25]  R. Fedele.  Envelope Solitons versus Solitons , 2002 .

[26]  H. Schamel,et al.  Solitary waves in the Madelung's fluid: Connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation , 2002 .

[27]  Alexey V. Porubov,et al.  Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer , 1996 .

[28]  Wen-Xiu Ma,et al.  EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS , 2008, 0812.4316.

[29]  Babak Parvin,et al.  Temporal behavior of an atom-cavity system in two distinct regimes , 2016 .

[30]  E. Ozturk,et al.  Nonlinear intersubband absorption and refractive index change in n-type δ-doped GaAs for different donor distributions , 2015 .

[31]  P. Shukla,et al.  Envelope solitons of nonlinear Schrödinger equation with an anti-cubic nonlinearity , 2002, nlin/0207054.

[32]  M. Dehghan,et al.  ANALYTICAL TREATMENT OF SOME PARTIAL DIFFERENTIAL EQUATIONS ARISING IN MATHEMATICAL PHYSICS BY USING THE Exp-FUNCTION METHOD , 2011 .

[33]  Anjan Biswas,et al.  1-soliton solution of the generalized Zakharov-Kuznetsov modified equal width equation , 2009, Appl. Math. Lett..

[34]  Abdul-Majid Wazwaz,et al.  Bright soliton solution to a generalized Burgers-KdV equation with time-dependent coefficients , 2010, Appl. Math. Comput..

[35]  H. Bart,et al.  Analytical solutions of the particle breakage equation by the Adomian decomposition and the variational iteration methods , 2015 .

[36]  J. Manafian,et al.  Optical solitons with Biswas-Milovic equation for Kerr law nonlinearity , 2015 .

[37]  J. Manafian,et al.  New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients , 2015 .

[38]  Jalil Manafian,et al.  On the complex structures of the Biswas-Milovic equation for power, parabolic and dual parabolic law nonlinearities , 2015 .

[39]  Andrea Klug,et al.  Nonlinear Physics For Beginners , 2016 .

[40]  Syed Tauseef Mohyud-Din,et al.  Analytical solutions Zakharov–Kuznetsov equations , 2013 .

[41]  A. Kundu.  Exact solutions to higher-order nonlinear equations through gauge transformation , 1987 .

[42]  M. Dehghan,et al.  Solving nonlinear fractional partial differential equations using the homotopy analysis method , 2010 .

[43]  J. Meng,et al.  New localized coherent structures to the (2+1)-dimensional breaking soliton equation [rapid communication] , 2004 .

[44]  Abdul-Majid Wazwaz,et al.  Partial differential equations : methods and applications , 2002 .