The discussion centers on the wave equation \(u_{tt} - u_{xx} = \sin(u)\) and its transformation using the substitution \(u(\xi) = u(x - ct)\). The transformation leads to the equation \((1 - c^2)u_{\xi\xi} = \sin(u)\), which is derived by applying the chain rule to express \(u_{tt}\) and \(u_{xx}\) in terms of \(\xi\). The correct formulation of the sine-Gordon equation is confirmed, emphasizing the importance of maintaining the correct sign in the equation.
PREREQUISITESMathematicians, physicists, and engineers working with wave phenomena, particularly those interested in nonlinear dynamics and the sine-Gordon equation.
If $u$ is a function of $\xi$, where $\xi = x-ct$, then the chain rule says that $u_t = \tfrac{\partial u}{\partial t} = \tfrac{du}{d\xi}\tfrac{\partial \xi}{\partial t} = -c\tfrac{du}{d\xi} = -cu_{\xi}.$ The same calculation, repeated, says that $u_{tt} = c^2u_{\xi\xi}.$ In the same way, $u_{xx} = u_{\xi\xi}.$ So the equation $u_{xx} - u_{tt} = \sin u$ becomes $(1-c^2)u_{\xi\xi} = \sin u.$ That is the standard form of the sine Gordon equation, but you seem to have changed a sign somewhere, writing $u_{tt} - u_{xx}$ rather than $u_{xx} - u_{tt}.$dwsmith said:Consider \(u_{tt} - u_{xx} = \sin(u)\). Let \(u(\xi) = u(x - ct)\).
How do we get \((1 - c^2)u_{\xi\xi} = \sin(u)\)?
\[
u_{\xi\xi} = u_{xx} - c^2u_{tt}
\]
Correct?