Exact solutions of cubic-quintic modified Korteweg-de-Vries equation (Q2066927)

The aim of this paper is to study a non-integrable modified Korteweg-de-Vries (mKdV) 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. Using the Painlevé analysis [\textit{R. M. Conte} and \textit{M. Musette}, The Painlevé handbook. Dordrecht: Springer (2008; Zbl 1153.34002)], the authors study the analytical structure of equation, obtained from the mKdV 3-5 equation by transition to a traveling wave variable, build its solution, expressed in terms of the Weierstrass elliptic function [\textit{A. V. Porubov}, J. Phys. A, Math. Gen. 26, No. 17, L\, 797-L\, 800 (1993; Zbl 0803.35132); \textit{R. Racke}, Appl. Anal. 58, No. 1--2, 85--100 (1995; Zbl 0832.35097)], classify exact and approximate partial solitary-wave and periodic solutions and plot the corresponding graphs. It is established that mKdV 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. This paper is organized as follows: The first section is an introduction to the subject. The second section deals with Painlevé analysis. The third and forth sections are devoted to periodic and soliton solutions. The fifth section deals with kink-shaped solution. The sixth section is devoted to approximate solution. For the entire collection see [Zbl 1471.74003].