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

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.