Let a non-linear PDE: $g(u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial t},\frac{\partial^2 u}{\partial x^2},\frac{\partial^2 u}{\partial t^2})=0$, where $u=u(x,t)$. The (real) variables $x,t$ are independent and $u$ is the dependent one (also real). So u: $\mathbb{R}^2\rightarrow \mathbb{D}$, where $\mathbb{D}$ is a subset of real numbers (or the whole set).
My questions have to do with the following:
Assume that out of the whole domain , we want to choose $t=f(x)$. This means that we are interested only in a curve $\mathbb{C}$ characterized by the previous relation. In this sense, I don't see why this is forbidden. On the other hand, by choosing it, we make $t$ a dependent parameter and I don't know if this is OK. Thus, can we do this?
Second part, assuming this transformation is fine, we want to re-write the PDE in terms only of $x$ (the only independent variable), that is $g'(u,\frac{\partial u}{\partial x}, \frac{\partial^2 u}{\partial x^2},)=0)$. This means that eventually we will end up with an ODE, as $x$ will be the only dependent variable, which means that we will be able to replace $\partial\rightarrow d$. To this end, I have tried to replace the t-derivatives with x-derivatives by using the chain rule:
$\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x}\frac{dx}{dt} \quad \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}\Big(\frac{dx}{dt}\Big)^2+\frac{\partial u}{\partial x}\frac{d^2x}{dt^2}$
In order to get the expression for $\frac{dx}{dt}$:
$dt=\frac{df}{dx}dx \quad \Rightarrow \quad \frac{dx}{dt}=\big(\frac{df}{dx}\big)^{-1}, \quad \frac{d^2x}{dt^2}=-\frac{d^2f}{dx^2}(\frac{df}{dx})^{-3}$
We may assume that $\frac{df}{dx}$ is non-zero for every value of x, to ensure smoothnees etc. However, this did not work. What I mean, we have a solution $u_0(x,t)$ for g, we make the transformations in order to get $g'$. Then the transformed $u_0=u_0(x,f(x))$ is not a solution of $g'$. Am I doing something wrong?
If we want to discuss specifically, I work with Sine-Gordon equation:
$\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}+\frac{m^2}{\beta}\sin\big(\beta u\big)=0$
As solution, I consider the one-soliton:
$u(x,t)=\frac{4}{\beta}\arctan\Big(\frac{m}{\sqrt{1-v^2}}(x-vt)+x_0\Big)$
$m, \beta, v, x_0$ are real and $|v|<1$. The transformation I considered: $t=x$