Abstract
We study a nonintegrable modified Korteweg-de-Vries equation containing a combination of third- and fifth-degree nonlinear terms that simulate waves in a three-layer fluid, as well as in spatially one-dimensional nonlinear-elastic deformable systems. It is established that this equation passes the Painlevé test in a weak form. After the traveling wave transformation, this equation reduces to a generalized Weierstrass elliptic function equation, the right side of which is determined by a sixth-order polynomial in the dependent variable. Determined by the structure of the polynomial roots, the general solution of the equation is expressed in terms of the Weierstrass elliptic function or its successive degenerations—rational functions depending on the exponential functions of the traveling wave variable or directly on traveling wave variable. The classification of exact solitary-wave and periodic solutions is carried out, and the ranges of parameters necessary for their physical feasibility are revealed. An approach is proposed for constructing approximate solitary-wave and periodic solutions to generalized Weierstrass elliptic equation with a polynomial right-hand side of high orders.
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The reported study was funded by RFBR, project number 20-01-00123.
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Zemlyanukhin, A.I., Bochkarev, A.V. (2020). Exact Solutions of Cubic-Quintic Modified Korteweg-de-Vries Equation. In: Altenbach, H., Eremeyev, V., Pavlov, I., Porubov, A. (eds) Nonlinear Wave Dynamics of Materials and Structures. Advanced Structured Materials, vol 122. Springer, Cham. https://doi.org/10.1007/978-3-030-38708-2_26
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DOI: https://doi.org/10.1007/978-3-030-38708-2_26
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