We consider the following nonlinear Schrödinger equation of derivative
type:
If
,
this equation is known as a standard derivative nonlinear Schrödinger equation
(DNLS), which is mass-critical and completely integrable. The equation above can be
considered as a generalized equation of DNLS while preserving mass-criticality
and Hamiltonian structure. For DNLS it is known that if the initial data
satisfies the
mass condition
,
the corresponding solution is global and bounded. In this paper we
first establish the mass condition on the equation above for general
, which corresponds
exactly to the
-mass
condition for DNLS, and then characterize it from the viewpoint of
potential well theory. We see that the mass-threshold value gives the
turning point in the structure of potential wells generated by solitons.
In particular, our results for DNLS give a characterization of both the
-mass
condition and algebraic solitons.
