Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term (Advances in Applied Mathematics and Mechanics, 2012)
In this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient ofnonlinear wave equation with cubic term by the theory and method ofplanar dynamical systems. Two critical values which can characterizethe scale of dissipation effect are obtained. If dissipation effectis not less than a certain critical value, the traveling wavesolutions appear as kink profile; while if it is less than thiscritical value, they appear as damped oscillatory. All expressionsof bounded traveling wave solutions are presented, including exactexpressions of bell and kink profile solitary wave solutions, aswell as approximate expressions of damped oscillatory solutions. Forapproximate damped oscillatory solution, using homogenizationprinciple, we give its error estimate by establishing the integralequation which reflects the relations between the exact andapproximate solutions. It can be seen that the error is aninfinitesimal decreasing in the exponential form.