Abstract
We have investigated the femtosecond soliton propagation in inhomogeneous fiber, which is described by the modified inhomogeneous Hirota equation with variable coefficient (MIH-vc). With the aid of AKNS method, corresponding Lax pair is constructed. By virtue of the Darboux transformation method and symbolic computation, the analytic one- and two-soliton solutions are explicitly obtained. Using obtained solutions, we graphically discuss the features of femtosecond solitons in modified inhomogeneous Hirota system by changing the profile of variable coefficients. We analyze various form of group velocity dispersion, third order dispersion and nonlinearity parameter for periodic amplification system, exponentially distributed system, parabolic solitons, periodic exponentially modulated system, which will be observable in the future experiments. These results are potentially useful in future experiments and soliton control for long-distance optical communication. Finally, the soliton solutions of the MIH-vc equation in double Wronskian form is constructed and further verified using the Wronskian technique by substitute in bilinear equations.
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References
Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical physics in dispersive dielectric fibers. Appl. Phys. Lett. 23, 142 (1973)
Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of picosecond pulse narrowing and solitons in Optical fibers. Phys. Rev. Lett. 45, 1095–1098 (1980)
Nakkeeran, K.: Optical solitons in erbium-doped fibres with higher-order effects and pum**. J. Phys. A Math. Gen. 33, 4377 (2000)
Hao, R.Y., Li, L., Li, Z.H., Zhou, G.S.: Exact multi-soliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients. Phys. Rev. E. 70, 066603 (2004)
Xue, Y.S., Tian, B., Ai, W.B., Qi, F.H., Guo, R., Qin, B.: Soliton interactions in a generalized inhomogeneous coupled Hirota–Maxwell–Bloch system. Nonlinear Dyn. 67, 2799–2806 (2012)
Kodama, Y., Hasegawa, A.: Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron. 23, 510 (1987)
Vicencio, R.A., Molina, M.I., Kivshar, Y.S.: Polarization instability, steering and switching of discrete vector solitons. Phys. Rev. E. 71, 056613 (2005)
Senthilnathan, K., Li, Q., Nakkeeran, K., Wai, P.K.A.: Robust pedestal-free pulse compression in cubic–quintic nonlinear media. Phys. Rev. A. 78, 033835 (2008)
Liu, W.J., Meng, X.H., Cai, K.J., Lu, X., Xu, T., Tian, B.: Analytic study on soliton-effect pulse compression in dispersion-shifted fibers with symbolic computation. J. Mod. Opt. 55, 1331–1344 (2008)
Ponomarenko, S.A., Agrawal, G.P.: Do soliton like self-similar waves exist in nonlinear optical media? Phys. Rev. Lett. 97, 013901 (2006)
Liu, W.J., Tian, B., Wang, P., Jiang, Y., Sun, K., Li, M., Qu, Q.X.: A new approach to the analytic soliton solutions for the variable-coefficient higher-order nonlinear Schrödinger model in inhomogeneous optical fibers. J. Mod. Opt. 57, 309–315 (2010)
Wu, X.F., Hua, G.S., Ma, Z.Y.: Evolution of optical solitary waves in a generalized nonlinear Schrödinger equation with variable coefficients. Nonlinear Dyn. 70, 2259–2267 (2012)
Zhu, H.P.: Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides. Nonlinear Dyn. 72, 873–882 (2013)
He, J.S., Tao, Y.S., Porsezian, K., Fokas, A.S.: Rogue wave management in an inhomogeneous nonlinear fiber with higher order effects. J. Nonli. Math. Phys. 20, 407–419 (2013)
Guo, R., Hao, H.Q.: Breathers and localized solitons for the Hirota–Maxwell–Bloch system on constant backgrounds in erbium doped fibers. Ann. Phys. 344, 10–16 (2014)
Xue, Y.S., Tian, B., Ai, W.B., Li, M., Wang, P.: Integrability and optical solitons in a generalized variable-coefficient coupled Hirota–Maxwell–Bloch system in fiber optics. Opt. Laser Technol. 48, 153–159 (2013)
Tian, H., Li, Z., Zhou, G.S.: Stable propagation of ultrashort optical pulses in modified higher-order nonlinear Schrödinger equation. Opt. Commun. 205, 221–226 (2002)
Ablowitz, M.J., Kaup, D.J., Newell, A.C., et al.: Nonlinear evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973)
Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)
Geng, X., Lv, Y.: Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation. Nonlinear Dyn. 69, 1621–1630 (2012)
Zhang, H.Q., Zhai, B.G., Wang, X.L.: Soliton and breather solutions of the modified nonlinear Schrödinger equation. Phys. Scr. 85, 015007 (2012)
Qi, F.H., Ju, H.M., Meng, X.H., Li, J.: Conservation laws and Darboux transformation for the coupled cubic-quintic nonlinear Schrödinger equations with variable coefficients in nonlinear optics. Nonlinear Dyn. doi:10.1007/s11071-014-1382-5
Yang, R.C., Li, L., Hao, R.Y., Li, Z.H., Zhou, G.S.: Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation. Phys. Rev. E 71, 036616 (2005)
Lü, X., Zhu, H.W., Meng, X.H., Yang, Z.C., Tian, B.: Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl. 336, 1305 (2007)
Dai, C.Q., Xu, Y.J., Chen, R.P., Zhang, J.F.: Self-similar optical beam in nonlinear waveguides. Eur. Phys. J. D 59, 457–461 (2010)
Zhang, J.L., Li, B.A., Wang, M.L.: The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients. Chaos Soliton. Fract. 39, 858–865 (2009)
Zheng, H., Wu, C., Wang, Z., Yu, H., Liu, S., Li, X.: Propagation characteristics of chirped soliton in periodic distributed amplification systems with variable coefficients. Optik 123, 818–822 (2012)
Mani Rajan, M.S., Mahalingam, A., Uthayakumar, A., Porsezian, K.: Observation of two soliton propagation in an erbium doped inhomogeneous lossy fiber with phase modulation. Commun. Nonlinear Sci. Numer. Simul. 18, 1410–1432 (2013)
Yang, R.C., Hao, R.Y., Li, L., Shi, X., Li, Z., Zhou, G.S.: Exact gray multi-soliton solutions for nonlinear Schrödinger equation with variable coefficients. Opt. Commun. 253, 177 (2005)
Jiang, L.H., Wu, H.Y.: Spatiotemporal self-similar waves for the (3+1)-dimensional inhomogeneous cubic–quintic nonlinear medium. Opt. Commun. 284, 2022–2026 (2011)
Dai, C.Q., Qin, Z.Y., Zheng, C.L.: Multi-soliton solutions to the modified nonlinear Schrödinger equation with variable coefficients in inhomogeneous fibers. Phys. Scr. 85, 045007 (2012)
Dai, C.Q., Xu, Y.J.: Spatial bright and dark similaritons on cnoidal wave backgrounds in 2D waveguides with different distributed transverse diffractions. Opt. Commun. 311, 216–221 (2013)
Serkin, V.N., Belyaeva, T.L.: Optimal control of optical soliton parameters: Part 1. The Lax representation in the problem of soliton management. Quantum Electron 31, 1007–1015 (2001)
Li, B., Chen, Y.: Symbolic computation and solitons of the nonlinear Schrödinger equation in inhomogeneous optical fiber media. Chaos Soliton. Fract. 33, 532 (2007)
Guo, R., Tian, B., Lü, X., Zhang, H.Q., Liu, W.J.: Darboux transformation and soliton solutions for the generalized coupled variable coefficient nonlinear Schrödinger Maxwell Bloch system with symbolic computation. Comput. Math. Math. Phys. 52, 565–577 (2012)
Fang, F., **ao, Y.: Stability of chirped bright and dark soliton-like solutions of the cubic complex Ginzburg–Landau equation with variable coefficients. Opt. Commun. 268, 305–310 (2006)
Wang, J., Li, L., Jia, S.: Exact chirped gray soliton solutions of the nonlinear Schrödinger equation with variable coefficients. Opt. Commun. 274, 223–230 (2007)
Freeman, N.C., Nimmo, J.J.: Soliton solutions of the Korteweg-de Vries and Kadomtsev–Petviashvili equations: the Wronskian technique. Phys. Lett. A 95, 1–3 (1983)
Tian, B., Gao, Y.T.: Symbolic-computation study of the perturbed nonlinear Schrödinger model in inhomogeneous optical fibers. Phys. Lett. A 342, 228 (2005)
Tian, B., Gao, Y.T.: Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: new transformation with burstons, brightons and symbolic computation. Phys. Lett. A 359, 241 (2006)
Tian, B., Gao, Y.T.: Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas. Phys. Lett. A 362, 283 (2007)
Lv, X., Zhu, H.W., Yao, Z.Z., Meng, X.H., Zhang, C., Zhang, C.Y., Tian, B.: Multisoliton solutions in terms of double Wronskian determinant for a generalized variable-coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications. Ann. Phys. 323, 1947–1955 (2008)
Sun, W.R., Tian, B., Jiang, Y.: Double-Wronskian solitons and rogue waves for the inhomogeneous nonlinear Schrödinger equation in an inhomogeneous plasma. Ann. Phys. 343, 215–227 (2014)
Freeman, N.C.: Soliton solutions of non-linear evolution equations. IMA J. Appl. Math. 32, 125–145 (1984)
Nimmo, J.J., Freeman, N.C.: A method of obtaining the \(N\)-soliton solution of the Boussinesq equation in terms of a wronskian. Phys. Lett. A 95, 4–6 (1983)
Freeman, N.C.: Soliton solutions of non-linear evolution equations. IMA J. Appl. Math. 32, 125–141 (1984)
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Mani Rajan, M.S., Mahalingam, A. Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction. Nonlinear Dyn 79, 2469–2484 (2015). https://doi.org/10.1007/s11071-014-1826-y
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DOI: https://doi.org/10.1007/s11071-014-1826-y