I have seen a lot of work has been done in the context of travelling wave. For example the work of McKenna and Chen in Journal of Differential Equations Volume 136, Issue 2, 20 May 1997, Pages 325-355. They are looking for the existence of travelling wave solutions of $$u''''+ c^2 u+ (u+1)_{+}-1=0 \text{ on } \mathbb{R}$$ where $c$ is called the wave speed and this comes from the substitution $u(x-ct)$. In physics, what is meant by the fact $c\to 0.$ What role does it play? Why is the concept of travelling wave important from the engineering point of view?
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1) Mathematical interest: This equation for $u$ is nonlinear ($u_+$ means $\max(u,0)$), so this is a mathematically interesting problem in the context of partial differential equations.
2) Engineering interest: A construction, such as a suspension bridge, that supports a travelling wave can exert large stress when the wave dissipates on the end points of the construction. In general, the design of the structure needs to be such that these waves will not propagate. The effect of a travelling wave has been studied in particular in the context of earthquake protection, see for example Traveling wave effect on the seismic response of a steel arch bridge subjected to near fault ground motions.
3) The role of $c$: this is the propagation velocity of the wave. It governs the strength of the nonlinearity.
answered Feb 8, 2018 at 11:51
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1$\begingroup$ In particular, for $c$ small, the nonlinearity is "weak" and approximation methods (such as series expansions) may be possible. Often in such problems, "small" is big enough to be interesting in practical applications. $\endgroup$ Commented Feb 8, 2018 at 13:23
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$\begingroup$ There is a following paper in "A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam" in Journal of Differential Equation 2006 where they claim that if $c\to 0$, probably the number of homoclinics increases. $\endgroup$– sadiazCommented Feb 8, 2018 at 15:49