# American Institute of Mathematical Sciences

March  2013, 12(2): 1075-1090. doi: 10.3934/cpaa.2013.12.1075

## Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution

 1 School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China, China, China

Received  August 2011 Revised  November 2012 Published  September 2012

In this paper, we apply the theory of planar dynamical systems to make a qualitative analysis to the traveling wave solutions of nonlinear Kakutani-Kawahara equation $u_t+uu_x+bu_{x x x}-a(u_t+uu_x)_x=0$ ($b>0, a\ge0$) and obtain the existent conditions of the bounded traveling wave solutions. In dispersion-dominant case, we find that the unique bounded traveling wave solution of this equation has not only oscillatory property but also damped property. Furthermore, according to the evolution of orbits in the global phase portraits, we present an approximate damped oscillatory solution for this equation by the undetermined coefficients method. Finally, by the idea of homogenization principles, we obtain an integral equation which reflects the relation between this approximate damped oscillatory solution and its exact solution, thereby having the error estimate. The error is an infinitesimal decreasing in exponential form. From the results in this paper, it can be seen that the damped oscillatory solution of Kakutani-Kawahara equation in dispersion-dominant case still remains some properties of solitary wave solution for KdV equation.
Citation: Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075
##### References:

show all references

##### References:
 [1] Jorge A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 787-804. doi: 10.3934/dcds.2004.10.787 [2] Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185 [3] Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307 [4] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [5] Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429 [6] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [7] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [8] Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061 [9] Zhaosheng Feng, Goong Chen, Sze-Bi Hsu. A qualitative study of the damped duffing equation and applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1097-1112. doi: 10.3934/dcdsb.2006.6.1097 [10] Guoshan Zhang, Shiwei Wang, Yiming Wang, Wanquan Liu. LS-SVM approximate solution for affine nonlinear systems with partially unknown functions. Journal of Industrial & Management Optimization, 2014, 10 (2) : 621-636. doi: 10.3934/jimo.2014.10.621 [11] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 [12] Qiusheng Qiu, Xinmin Yang. Scalarization of approximate solution for vector equilibrium problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 143-151. doi: 10.3934/jimo.2013.9.143 [13] Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013 [14] Olivier Goubet. Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 503-530. doi: 10.3934/dcds.1997.3.503 [15] Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure & Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042 [16] Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control & Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1 [17] T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277 [18] Bassam Kojok. Global existence for a forced dispersive dissipative equation via the I-method. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1401-1419. doi: 10.3934/cpaa.2009.8.1401 [19] Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1303-1309. doi: 10.3934/jimo.2016.12.1303 [20] Nguyen Ba Minh, Pham Huu Sach. Strong vector equilibrium problems with LSC approximate solution mappings. Journal of Industrial & Management Optimization, 2020, 16 (2) : 511-529. doi: 10.3934/jimo.2018165

2018 Impact Factor: 0.925