BOUNDED TRAVELING WAVE SOLUTIONS OF VARIANT BOUSSINESQ EQUATION WITH A DISSIPATION TERM AND DISSIPATION EFFECT

• Published in: Acta mathematica scientia Vol. 34; no. 3; pp. 941 - 959
• Main Author: 张卫国 刘强 李正明 李想
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.05.2014
• Subjects: 35G20; 35Q35; 35Q51; bounded traveling wave solution; Boussinesq equations; Dissipation; dissipation effect; Dynamical systems; error estimate; Evolution; Mathematical analysis; Qualitative analysis; shape analysis; Traveling waves; Variant Boussinesq equation with dissipation term; 变形Boussinesq方程; 平面动力系统; 形状演化; 散热效果; 耗散效应; 耗散项; 行波解; 阻尼振荡波; error estimate; bounded traveling wave solution; 35G20; 35Q35; Variant Boussinesq equation with dissipation term; shape analysis; 35Q51; dissipation effect
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1016/S0252-9602(14)60061-8

This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r^＊ which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if ｜r｜≥ r^＊; while they appear as damped oscillatory waves if ｜r｜ 〈 r^＊. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u（ξ） in the global phase portraits, and then obtain the approximate solutions （u（ξ）, H（ξ））. Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.