# The First Integral Method and its Application for Deriving the Exact Solutions of a Higher-Order Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

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The objective of this article is to apply the first integral method to construct the exact solutions for a higher-order dispersive cubic-quintic nonlinear Schrödinger equation describing the propagation of extremely short pulses. Using a simple transformation, this equation can be reduced to a nonlinear ordinary differential equation (ODE). Various solutions of the ODE are obtained by using the first integral method. Further results are obtained by using a direct method. A comparison between our results and the well-known results is given.

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First integral method higher-order dispersive cubic-quintic nonlinear Schrödinger equation exact solutions division theorem## Preview

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## References

- 1.E. J. Parkes and B. R. Duffy, “An automated tanh-function method for finding solitary wave solutions to non-linear evolution equation,”
*Comput. Phys. Commun.,***98**, 288–300 (1996).MATHCrossRefGoogle Scholar - 2.E. G. Fan, “Extended tanh-function method and its applications to nonlinear equations,”
*Phys. Lett. A*,**277**, 212–218 (2000).MATHMathSciNetCrossRefGoogle Scholar - 3.Z. Y. Yan, “New explicit traveling wave solutions for two new integrable coupled nonlinear evolution equations,”
*Phys. Lett. A*,**292**, 100–106 (2001).MATHMathSciNetCrossRefGoogle Scholar - 4.S. Liu and Z. Fu, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,”
*Phys. Lett. A*,**289**, 69–74 (2001).MATHMathSciNetCrossRefGoogle Scholar - 5.Z. Y. Yan, “Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey–Stewartson-type equation via a new method,”
*Chaos Solitons Fract.*,**18**, 299–309 (2003).MATHCrossRefGoogle Scholar - 6.Z. T. Fu and S. K. Liu, “A new approach to solve nonlinear wave equations,”
*Commun. Theor. Phys.*,**39**, 27–30 (2003).MATHMathSciNetCrossRefGoogle Scholar - 7.M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,”
*Phys. Lett. A,***199**, 169–172 (1995).MathSciNetCrossRefGoogle Scholar - 8.M. L. Wang, “Exact solutions for a compound KdV–Burgers equation,”
*Phys. Lett. A*,**213**, 279–287 (1996).MATHMathSciNetCrossRefGoogle Scholar - 9.M. L. Wang, Y. B. Zhou, and Z. B. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,”
*Phys. Lett. A*,**216**, 67–75 (1996).MATHCrossRefGoogle Scholar - 10.E. M. E. Zayed and A. H. Arnous, “DNA dynamics studied using the homogeneous balance method,”
*Chin. Phys. Lett.*,**29**, 080203–080205 (2012).CrossRefGoogle Scholar - 11.M. L. Wang and Y. B. Zhou, “The periodic wave solutions for the Klein–Gordon–Schrödinger equations,”
*Phys. Lett. A*,**318**, 84–92 (2003).MATHMathSciNetCrossRefGoogle Scholar - 12.Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solution and solitary wave solutions for a class of nonlinear partial differential equations,”
*Phys. Lett. A*,**323**, 77–88 (2004).MATHMathSciNetCrossRefGoogle Scholar - 13.M. L. Wang and X. Z. Li, “Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations,”
*Phys. Lett. A*,**343**, 48–54 (2005).MATHMathSciNetCrossRefGoogle Scholar - 14.M. L. Wang and X. Z. Li, “Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation,”
*Chaos Soliton Fract.,***24**, 1257–1268 (2005).MATHCrossRefGoogle Scholar - 15.J. L. Zhang, M. L. Wang, and X. Z. Li, “The subsidiary ordinary differential equations and the exact solutions for the higher order dispersive nonlinear Schrödinger equation,”
*Phys. Lett. A*,**357**, 188–195 (2006).MATHCrossRefGoogle Scholar - 16.M. L. Wang, J. L. Zhang, and X. Z. Li, “Various exact solutions for the nonlinear Schrödinger equation with two nonlinear terms,”
*Chaos Soliton Fract*.,**31**, 594–601 (2007).MATHMathSciNetCrossRefGoogle Scholar - 17.X. Z. Li and M. L. Wang, “A sub-ODE method for finding exact solutions of a generalized KdV and mKdV equation with highorder nonlinear terms,”
*Phys. Lett. A*,**361**, 115–118 (2007).MATHMathSciNetCrossRefGoogle Scholar - 18.M. L. Wang, X. Z. Li, and J. L. Zhang, “Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation,”
*Phys. Lett. A*,**363**, 96–101 (2007).MATHMathSciNetCrossRefGoogle Scholar - 19.E. Yomba, “The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation,”
*Chaos Soliton Fract*.,**27**, 187–196 (2006).MATHMathSciNetCrossRefGoogle Scholar - 20.X. Y. Li, X. Z. Li, and M. L. Wang, “Extended F-expansion method and periodic wave solutions for Klein–Gordon–Schrödinger equations,”
*Commun. Theor. Phys*.,**45**, 9–14 (2006).MATHCrossRefGoogle Scholar - 21.J. H. He and X. H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons Fract*.,**30**, 700–708 (2006).MATHMathSciNetCrossRefGoogle Scholar - 22.E. M. E. Zayed and M. A. M. Abdelaziz, “Exact solutions for the Schrodiner equation with variable coefficients using a generalized extended tanh-function, the sine-cosine and the Exp-function methods,”
*Appl. Math. Comput*.,**218**, 2259–2268 (2011).MATHMathSciNetCrossRefGoogle Scholar - 23.S. D. Zhu, “Exp-function method for the discrete mKdV lattice,”
*Int. J. Nonlinear Sci. Numer. Simul*.,**8**, 465–469 (2007).Google Scholar - 24.M. Wang, J. Zhang, and X. Li, “The (
*G*′**/***G*)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics,”*Phys. Lett. A*,**372**, 417–423 (2008).MATHMathSciNetCrossRefGoogle Scholar - 25.S. Zhang, J. L. Tong, and W. Wang, “A generalized (
*G*′**/***G*) -Expansion method for the mKdV equation with variable coefficients,”*Phys. Lett. A*,**372,**2254–2257 (2008).MATHMathSciNetCrossRefGoogle Scholar - 26.E. M. E. Zayed and K. A. Gepreel,” The (
*G*′**/***G*) -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics,”*J. Math. Phys*.,**50**, 13502–13513 (2009).MathSciNetCrossRefGoogle Scholar - 27.E. M. E. Zayed and K. A. Gepreel, “Some application of the (
*G*′**/***G*) -expansion method to nonlinear partial differential equations,”*Appl. Math. Comput*.,**212**, 1–13 (2009).MATHMathSciNetCrossRefGoogle Scholar - 28.E. M. E. Zayed, “New traveling wave solution for higher dimensional nonlinear evolution equations using a generalized (
*G*′**/***G*) - expansion method,”*J. Phys. A: Math. Theor*.,**42**, 195202–195214 (2009).MathSciNetCrossRefGoogle Scholar - 29.E. M. E. Zayed, “The (
*G*′**/***G*) -expansion method and its applications to some nonlinear evolution equations in the mathematical physics,”*J. Appl. Math. Comput.*,**30**, 89–103 (2009).MATHMathSciNetCrossRefGoogle Scholar - 30.E. M. E. Zayed and M. A. M. Abdelaziz, “Traveling wave solution for the Burgers equation and the KdV equation with variable coefficients using a generalized (
*G*′**/***G*) -expansion method,”*Z. Naturforsch.*,**65a**, 1065–1070 (2010).Google Scholar - 31.Z. S. Feng, “The first integral method to study the Burgers–KdV equation,”
*J. Phys. A: Math. Gen*.,**35**, 343–349 (2002).MATHCrossRefGoogle Scholar - 32.B. Zheng, “Traveling wave solution for some nonlinear evolution equations by the first integral method,”
*WSEAS Trans. Math*.,**8**, 249–258 (2011).Google Scholar - 33.N. Taghizadeh, M. Mirzazadeh, and F. Farahrooz, “Exact solutions of the nonlinear Schrödinger equation by the first integral method,”
*J. Math. Anal. Appl*.,**374**, 549–553 (2011).MATHMathSciNetCrossRefGoogle Scholar - 34.P. Sharma and O. Y. Kushel, “The first integral method for Huxley equation,”
*Int. J. Nonlinear Sci*.,**10**, 46–52 (2010).MATHMathSciNetGoogle Scholar - 35.A. Bekir and O. Unsal, “Analytical treatment of nonlinear evolution equations using the first integral method,”
*Pramana J. Phys*.,**79**, 3–17 (2012).CrossRefGoogle Scholar - 36.B. Lu, H. Zhang, and F. Xie, “Traveling wave solutions of nonlinear partial equations by using the first integral method,”
*Appl. Math. Comput*.,**216**, 1329–1336 (2010).MATHMathSciNetCrossRefGoogle Scholar - 37.S. Zhu, “Exact solutions of the high-order dispersive cubic quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method,”
*Chaos Soliton Fract*.,**34**, 1608–1612 (2007).MATHCrossRefGoogle Scholar - 38.J. L. Zhang, M. L. Wang, and X. Z. Li, “The subsidiary ordinary differential equations and the exact higher-order dispersive nonlinear Schrödinger equations,”
*Phys. Lett. A*.,**357**, 188–195 (2006).MATHCrossRefGoogle Scholar - 39.S. L. Palacios and J. M. Ferandez-Diaz, “Black optical solitons for media with parabolic nonlinearity law in the presence of fourth order dispersion,”
*Opt. Commun*.,**178**, 457–460 (2000).CrossRefGoogle Scholar - 40.S. Tanev and D. I. Pushkarov, “Solitary wave propagation and bistability in the normal dispersion region of highly nonlinear optical fibers and waveguides,”
*Opt. Commun*.,**141**, 322–328 (1997).CrossRefGoogle Scholar - 41.A. G. Shagalov, “Modulational instability of nonlinear waves in the range of zero dispersion,”
*Phys. Lett. A*,**239**, 41–45 (1998).MATHMathSciNetCrossRefGoogle Scholar - 42.Z. S. Feng, “The first integral method to study the Burgers–KdV equation,”
*J. Phys. A: Math. Gen*.,**35**, 343–349 (2002).MATHCrossRefGoogle Scholar - 43.W. X. Ma and B. Fuchssteiner, “Explicit and exact solutions to Kolmogorov–Petrovskii–Piskunov equation,”
*Int. J. Nonlinear Mech*.**31**, 329–338 (1996).MATHMathSciNetCrossRefGoogle Scholar - 44.Sirendaoreji, “A new auxiliary equation a nd exact traveling wave solutions of nonlinear equations,”
*Phys. Lett. A*,**356**, 124–130 (2006).MATHCrossRefGoogle Scholar - 45.Sirendaoreji, “Exact traveling wave solutions for four forms of nonlinear Klein–Gordon equations,”
*Phys. Lett. A*,**363**, 440–447 (2007).MATHMathSciNetCrossRefGoogle Scholar - 46.Sirendaoreji, “Auxiliary equation method and new solutions of Klein–Gordon equations,”
*Chaos, Soliton Fract*.,**31**, 943–950 (2007).MATHMathSciNetCrossRefGoogle Scholar - 47.E. M. E. Zayed, “A further improved (
*G*′**/***G*) -expansion method and the extended tanh method for finding exact solutions of nonlinear PDEs,”*WSEAS, Trans. Math*.,**10**, 56–64 (2011).MATHGoogle Scholar - 48.E. M. E. Zayed, “Observations on a further improved (
*G*′**/***G*) -expansion method and the extended tanh method for finding exact solutions of nonlinear PDEs,”*J. Appl. Math. Inform.*,**30**, 253–264 (2012).MATHMathSciNetGoogle Scholar

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