Abstract
We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV-Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical valuesλ1andλ2which can characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipation coefficientα≥λ1, it appears as a monotone kink profile solitary wave solution; that if0<α<λ1, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.
Highlights
Generalized KdV equation with dissipation term ut bupux − αuxx uxxx 01.1 is the physical model describing the long-wave propagating in nonlinear media with dispersion-dissipation 1, where α ≥ 0, b is any real number and p is any positive integer
We righttraveling wave of the equation, a bounded traveling wave solution appears as a monotones kink profile solitary wave solution if dissipation coefficient α ≥ λ1, while it appears as a damped oscillatory wave if 0 < α < λ1; for the left-traveling wave of the equation, a bounded traveling wave solution appears as a monotones kink profile solitary wave solution if dissipation coefficient α ≥ λ2, while it appears as a damped oscillatory wave if 0 < α < λ2
KdV1.1, if dissipation effect is large, namely, α ≥ λ1, the traveling wave solution of 1.1 appears as a monotone kink profile solitary wave; while if dissipation effect is small, namely, 0 < α < λ1, it appears as a damped oscillatory wave
Summary
We focus on studying approximate solutions of damped oscillatory solutions of generalized KdVBurgers equation and their error estimates. We obtain that for the right-traveling wave solution if dissipation coefficient α ≥ λ1, it appears as a monotone kink profile solitary wave solution; that if 0 < α < λ1, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions.
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