Behavior of a Solution of a Nonlinear ODE

Question

In my work, I encountered the following equation: $$ (a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x). $$ I would like to know as much as possible about the solution. More particularly, I would like to say that if $a(x)$ is bounded as $x\rightarrow \infty$, then $a(x)\rightarrow 0$. The same as $x\rightarrow -\infty$. If more information can be obtained, I would happily welcome it.

Answer 1

There are really two separate cases: $$\eqalign{a' + 1 &= +\sqrt{1 - k^2 a^2}\cr a' + 1 &= -\sqrt{1 - k^2 a^2}\cr} $$ You could switch continuously from one to the other only on the curve $1 - k^2 a^2 = 0$ (but at almost all points on that curve the trajectories will be either entering or leaving the forbidden region $1 - k^2 a^2 < 0$).

Let's consider the $+$ case, with an initial condition $a(x_0) = a_0$ where $x_0 > 0$ and $0 < a_0 k(x_0) < 1$. For $x > x_0$ we will have $0 < a(x) < a_0 < 1/k(x_0)$, and $$ 1 - \dfrac{k^2 a^2}{2} > \sqrt{1-k^2 a^2} > 1 - k^2 a^2 $$ so $$-\dfrac{k^2}{2} > \dfrac{a'}{a^2} = - \left(\frac{1}{a}\right)' > -k^2$$ Thus for $x > x_0$ $$ - 2 \tanh(x) + 2 \tanh(x_0) > \dfrac{1}{a(x_0)} - \dfrac{1}{a(x)} > -4 \tanh(x) + 4 \tanh(x_0) $$ Of course, as $x \to +\infty$ we have $\tanh(x) \to 1$, and we find that the limit of $a(x)$ must be nonzero.