We give a sufficient condition for global existence of the solutions to a generalized
derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The
variational argument is applicable to a cubic derivative nonlinear Schrödinger equation
(DNLS). For (DNLS), Wu (2015) proved that the solution with the initial data
is global
if
by
the sharp Gagliardo–Nirenberg inequality. The variational argument gives us
another proof of the global existence for (DNLS). Moreover, by the variational
argument, we can show that the solution to (DNLS) is global if the initial data
satisfies
and the
momentum
is negative.
Keywords
variational structure, generalized derivative nonlinear
Schrödinger equation, global existence