Is the following equation well known in MATHEMATICAL PHYSICS?
$$\alpha u_{tt} - {\rm i}u_t -u_{xx}-u^3 =0.$$ Here $u = u(t,x)$ is a complex valued function and $0<\alpha <<1.$ The only well known equation having the linear part of the above one is called the Klein-Gordon equation, but with non-linear term of the form $f(|u|^2)u.$
Thank you in advance for any relevant reference.