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Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy

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  • Kudryashov, Nikolay A.

Abstract

The traveling wave reduction of the Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found. The Lax pair is used to obtain the first integrals of the traveling wave reduction for the Korteweg-de Vries hierarchy. Exact formulas for the first integrals of the hierarchy are given. The first three members of the hierarchy are considered in more detail. These first integrals are the examples of the integrable nonlinear differential equations with the Painlevé property. Exact solutions in the form of the soliton for the traveling wave reduction of the Korteweg-de Vries hierarchy and its first integrals are presented.

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  • 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.
    • Handle: RePEc:eee:apmaco:v:350:y:2019:i:c:p:323-330
    DOI: 10.1016/j.amc.2019.01.034
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    References listed on IDEAS

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    1. 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.
    2. Polyanin, Andrei D., 2019. "Functional separable solutions of nonlinear reaction–diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 282-292.
    3. Kudryashov, Nikolai A., 2005. "Fuchs indices and the first integrals of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 591-603.
    4. Khalid, Muhammad Zeeshan & Zubair, Muhammad & Ali, Majid, 2019. "An analytical method for the solution of two phase Stefan problem in cylindrical geometry," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 295-308.
    5. Shen, Yu-Jia & Gao, Yi-Tian & Meng, Gao-Qing & Qin, Yi & Yu, Xin, 2016. "Improved Bell-polynomial procedure for the higher-order Korteweg–de Vries equations in fluid dynamics," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 403-413.
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    Cited by:

    1. Kudryashov, Nikolay A. & Lavrova, Sofia F., 2024. "Traveling wave solutions of the derivative nonlinear Schrödinger hierarchy," Applied Mathematics and Computation, Elsevier, vol. 477(C).
    2. Kudryashov, Nikolay A., 2020. "Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    3. Guan, Xue & Liu, Wenjun & Zhou, Qin & Biswas, Anjan, 2020. "Some lump solutions for a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    4. Kudryashov, Nikolay A., 2024. "Solitons of the complex modified Korteweg–de Vries hierarchy," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    5. Juan Luis García Guirao & Haci Mehmet Baskonus & Ajay Kumar, 2020. "Regarding New Wave Patterns of the Newly Extended Nonlinear (2+1)-Dimensional Boussinesq Equation with Fourth Order," Mathematics, MDPI, vol. 8(3), pages 1-9, March.
    6. AlQahtani, Salman A. & Alngar, Mohamed E.M. & Shohib, Reham M.A. & Pathak, Pranavkumar, 2023. "Highly dispersive embedded solitons with quadratic χ(2) and cubic χ(3) non-linear susceptibilities having multiplicative white noise via Itô calculus," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    7. Kudryashov, Nikolay A., 2020. "Optical solitons of model with integrable equation for wave packet envelope," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    8. Triki, Houria & Choudhuri, Amitava & Zhou, Qin & Biswas, Anjan & Alshomrani, Ali Saleh, 2020. "Nonautonomous matter wave bright solitons in a quasi-1D Bose-Einstein condensate system with contact repulsion and dipole-dipole attraction," Applied Mathematics and Computation, Elsevier, vol. 371(C).

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