[1] Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion[J]. Applied Physics Letters, 23, 142-144(1973).

[2] [M]. Nonlinear fiber optics & application of nonlinear fiber optics, 94-104(2002).

     Agrawal G P, Agrawal G P[M]. 非线性光纤光学原理及应用, 94-104(2002).

[3] Kruglov V I, Peacock A C, Harvey J D. Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients[J]. Physical Review E, 71, 056619(2005).

[4] Yang J W, Gao Y T, Feng Y J et al. Solitons and dromion-like structures in an inhomogeneous optical fiber[J]. Nonlinear Dynamics, 87, 851-862(2017).

[5] Kong D F, Jia D F, Feng D J et al. Soliton self-frequency shift in optical fibers[J]. Laser & Optoelectronics Progress, 55, 101902(2018).

[6] Wang L, Yang R C, Jia H P et al. Periodically lumped amplification and recovery of soliton in dispersion-decreasing optic fiber link[J]. Acta Optica Sinica, 37, 0619001(2017).

[7] Veselago V G. The electrodynamics of substances with simultaneously negative values of ε and μ[J]. Soviet Physics Uspekhi, 10, 509-514(1968).

[8] Lee S H, Park C M, Seo Y M et al. Reversed Doppler effect in double negative metamaterials[J]. Physical Review B, 81, 241102(2010).

[9] Shalaev V M. Optical negative-index metamaterials[J]. Nature Photonics, 1, 41-48(2007).

[10] Yang R C, Shadrivov I V. Double-nonlinear metamaterials[J]. Applied Physics Letters, 97, 231114(2010).

[11] Fan K B, Hwang H Y, Liu M K et al. Nonlinear terahertz metamaterials via field-enhanced carrier dynamics in GaAs[J]. Physical Review Letters, 110, 217404(2013).

[12] Gao F, Yang R C, Jia H P et al. Exact solution of spatial chirped dark soliton in metamaterials and its propagation characteristics[J]. Acta Optica Sinica, 39, 0219001(2019).

[13] Kozyrev A B, Shadrivov I V, Kivshar Y S. Soliton generation in active nonlinear metamaterials[J]. Applied Physics Letters, 104, 084105(2014).

[14] English L Q, Wheeler S G, Shen Y et al. Backward-wave propagation and discrete solitons in a left-handed electrical lattice[J]. Physics Letters A, 375, 1242-1248(2011).

[15] Lazarides N, Tsironis G P. Publisher's note: coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials [Phys. Rev. E71, 036614 (2005)][J]. Physical Review E, 71, 049903(2005).

[16] Scalora M, Syrchin M S, Akozbek N et al. Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials[J]. Physical Review Letters, 95, 013902(2005).

[17] Wen S C, Xiang Y J, Dai X Y et al. Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials[J]. Physical Review A, 75, 033815(2007).

[18] Li P G, Yang R C, Xu Z Y. Gray solitary-wave solutions in nonlinear negative-index materials[J]. Physical Review E, 82, 046603(2010).

[19] Sharma V K, Goyal A, Raju T S et al. Periodic and solitary wave solutions for ultrashort pulses in negative-index materials[J]. Journal of Modern Optics, 60, 836-840(2013).

[20] Yang R C, Min X M, Tian J P et al. New types of exact quasi-soliton solutions in metamaterials[J]. Physica Scripta, 91, 025201(2016).

[21] Yang R C, Zhang Y. Exact combined solitary wave solutions in nonlinear metamaterials[J]. Journal of the Optical Society of America B, 28, 123-127(2011).

[22] Triki H, Zhou Q, Moshokoa S P et al. Novel singular solitons in optical metamaterials for self-steepening effect[J]. Optik, 154, 545-550(2018).

[23] Biswas A, Ekici M, Sonmezoglu A et al. Chirped solitons in optical metamaterials with parabolic law nonlinearity by extended trial function method[J]. Optik, 160, 92-99(2018).

[24] Daoui A K, Triki H, Biswas A et al. Chirped bright and double-kinked quasi-solitons in optical metamaterials with self-steepening nonlinearity[J]. Journal of Modern Optics, 66, 192-199(2019).

[25] Daoui A K, Azzouzi F, Triki H et al. Propagation of chirped gray optical dips in nonlinear metamaterials[J]. Optics Communications, 430, 461-466(2019).

[26] Li X Z, Wang M L. A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms[J]. Physics Letters A, 361, 115-118(2007).

[27] Serge D Y, Justin M, Betchewe G et al. Optical chirped soliton in metamaterials[J]. Nonlinear Dynamics, 90, 13-18(2017).

[28] Dong J, Jin Y J, Chen C et al. The transmission properties of exact bright-like and dark-like solitons in metamaterials[J]. Acta Sinica Quantum Optica, 19, 162-170(2013).

[29] Ferrando A. Nonlinear plasmonic amplification via dissipative soliton-plasmon resonances[J]. Physical Review A, 95, 013816(2017).

[30] Zhang Q, Tan C H, Huang G X. Lossless Airy surface polaritons in a metamaterial via active Raman gain[J]. Scientific Reports, 6, 21143(2016).