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?
The equation being solved is utt - uxx = sin(u). This is a partial differential equation (PDE) that involves the second derivative of u with respect to time (t) and space (x).
u(ξ) represents the solution to the given PDE at a specific point in space and time. It is a function of the variables ξ, which can be any point in the spatial domain.
This equation is typically solved using numerical methods, such as finite difference or finite element methods. These methods involve discretizing the equation into a system of algebraic equations and solving them iteratively to approximate the solution at each time step.
The sine function in this equation represents a forcing term or external input that affects the behavior of u. This can represent a variety of physical phenomena, such as a periodic external force or a nonlinear interaction between different parts of the system.
This equation has many potential applications in physics and engineering, such as modeling wave propagation, heat transfer, and fluid dynamics. It can also be used to study nonlinear behavior and stability in various systems.