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Optical solitons of model with integrable equation for wave packet envelope

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  • Kudryashov, Nikolay A.

Abstract

We consider the nonlinear fourth-order partial differential equation that can be used for describing solitary waves in nonlinear optics. The Cauchy problem for this equation is not solved by the inverse scattering transform. However we demonstrate that nonlinear ordinary differential equation for description of the wave packet envelope possesses the Painlevé property and is integrable. The Lax pair to this nonlinear ordinary differential equation is presented. Using the determinant for the Lax pair matrix, we find the first integrals of a nonlinear ordinary differential equation. The general solution of the fourth-order nonlinear differential equation is given via the ultraelliptic integrals. Special cases of exact solutions for the fourth-order equation are expressed in terms of the Jacobi elliptic sine. Optical solitons of the original partial differential equation are found.

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  • Kudryashov, Nikolay A., 2020. "Optical solitons of model with integrable equation for wave packet envelope," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    • Handle: RePEc:eee:chsofr:v:141:y:2020:i:c:s0960077920307207
    DOI: 10.1016/j.chaos.2020.110325
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    References listed on IDEAS

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    1. Azzouzi, F. & Triki, H. & Mezghiche, K. & El Akrmi, A., 2009. "Solitary wave solutions for high dispersive cubic-quintic nonlinear Schrödinger equation," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1304-1307.
    2. Kudryashov, Nikolay A., 2020. "Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    3. Kudryashov, Nikolay A., 2019. "Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 323-330.
    4. Kudryashov, Nikolay A., 2020. "First integrals and general solution of the complex Ginzburg-Landau equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    5. Li, Pengfei & Malomed, Boris A. & Mihalache, Dumitru, 2020. "Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    6. Zayed, E.M.E. & Alurrfi, K.A.E., 2016. "Extended auxiliary equation method and its applications for finding the exact solutions for a class of nonlinear Schrödinger-type equations," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 111-131.
    7. Kudryashov, Nikolay A., 2019. "Exact solutions of the equation for surface waves in a convecting fluid," Applied Mathematics and Computation, Elsevier, vol. 344, pages 97-106.
    8. Kudryashov, Nikolay A., 2020. "Highly dispersive optical solitons of equation with various polynomial nonlinearity law," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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    Cited by:

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