Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8399Research ArticleModeling the Track Formation in Amorphous Iron Alloys Exposed to High-Energy Heavy IonsKudryashovN Anakudryashov@mephi.ruSinelshchikovD Idisinelshchikov@mephi.ruNational Research Nuclear University MEPHI15022014239439808092016Copyright © 2014,2014Nonlinear waves in liquid with gas bubbles are investigated taken into account liquid viscosity and compressibility and inter phase heat transfer. The nonlinear differential equation for long weakly nonlinear waves is obtained with the help of the reductive perturbation method. At the derivation of the equation higher order corrections in the asymptotic expansion are taken into account. This equation is the generalization of the Burgers equation and describes nonlinear waves in a liquid with gas bubbles in the case of dissipation main influence. The normal form is constructed for the equation with the help of the near-identity transformations. It is shown that the normal form equation is integrable under certain condition on parameters. In this case the equation for nonlinear waves is the second member of the Burgers hierarchy. Exact solution in the form of kink is obtained in the general case. Dependence of this solution on physical parameters is investigated. It is shown that the amplitude of this exact solution decreases when the bubbles radius in the unperturbed state and the liquid viscosity increase.liquid with gas bubblesnonlinear wavesnonlinear evolution equationsexact solutionsnormal formжидкость с пузырьками газанелинейные волнынелинейные эволюционные уравненияточные решениянормальная форма[Nakoryakov V. E., Pokusaev B. G., Shreiber I. G. Wave Propagation in Gas-Liquid Media. - Boca Raton: CRC Press, 1993.][Nigmatulin R. I. Dynamics of Multiphase Media, Part 2. - New York: Taylor & Francis, 1990.][Wijngaarden L. V. One-Dimensional Flow of Liquids Containing Small Gas Bubbles // Annu. Rev. Fluid Mech. - 1972. - Vol. 4. - Pp. 369-396.][Nakoryakov V. E., Sobolev V. V., Shreiber I. R. Longwave Perturbations in a GasLiquid Mixture // Fluid Dyn. - 1972. - Vol. 7, No 5. - Pp. 763-768.][Kudryashov N. A., Sinelshchikov D. I. Nonlinear Waves in Bubbly Liquids with Consideration for Viscosity and Heat Transfer // Phys. Lett. A. - 2010. - Vol. 374, No 19-20. - Pp. 2011-2016.][Kudryashov N. A., Sinelshchikov D. I. An Extended Equation for the Description of Nonlinear Waves in a Liquid with Gas Bubbles // Wave Mot. - 2013. - Vol. 50, No 3. - Pp. 351-362.][Kudryashov N. A. Simplest Equation Method to Look for Exact Solutions of Nonlinear Differential Equations // Chaos Soliton Fract. - 2005. - Vol. 24, No 5. - Pp. 1217-1231.]