I'm having a hard time trying to understand the following derivation, from G.B. Whitham's book Linear and Nonlinear Waves, 1999 (Chap. 13, p. 466).
The equations for $A$ are: $$ A_x + 2\eta\eta_x=0 \qquad A_t + \eta\eta_x=0 $$ so I would integrate the equation on the left to: $$ A = -\eta^2 $$ and I would rewrite the equation on the right to: $$ A_t = \eta\eta_t $$ which leads to $$ A=\frac12 \eta^2. $$ So how can these two equations be compatible? A similar doubt holds for $B$.
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$\begingroup$We want the two equations to be compatible, i.e. the coefficients of the first order terms to be equal (not necessarily zero). Replacing all $t$ derivatives by minus the $x$ derivatives in the first order terms, \begin{aligned} A_x + 2\eta\eta_x &= A_t + \eta\eta_x \\ &= -A_x + \eta\eta_x \, , \end{aligned} we get $2A_x + \eta\eta_x = 0$. Integrating with respect to $x$ gives $A = -\frac{1}{4}\eta^2$, since $A$ is a function of $\eta$ and its $x$ derivatives only.
The derivation for $B$ is similar.
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