Power type

A nonlinear function ${\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ is of power type with exponent ${\displaystyle p>1}$ if one has the bounds

${\displaystyle |F(u)|\leq C|u|^{p}}$

(so in particular ${\displaystyle F(0)=\nabla F(0)=0}$) and

${\displaystyle |F(u)-F(v)|\leq C|u-v|(|u|^{p-1}+|v|^{p-1})}$

for all ${\displaystyle u,v\in \mathbb {R} ^{n}}$ and some constant ${\displaystyle C>0}$. Note that the first bound is a special case of the second once one assumes that ${\displaystyle F(0)=0}$. If F is continuously differentiable, then the second bound is equivalent to the bound

${\displaystyle |\nabla F(u)|\leq C|u|^{p-1}}$

(possibly for a slightly different value of C), thanks to the fundamental theorem of calculus identity

${\displaystyle F(u)-F(v)=\int _{0}^{1}(u-v)\cdot \nabla F((1-t)u+tv)\ dt.}$

The model example of a power type nonlinearity is the pure power nonlinearity ${\displaystyle F(u)=\pm |u|^{p-1}u}$ (for either real or complex u). If p is an integer, any function F which is a homogeneous polynomial of degree p in u and (in the complex case) ${\displaystyle {\overline {u}}}$ also qualifies as a power type nonlinearity. As a rule of thumb, local well-posedness results which hold for pure power nonlinearities, also hold for power type nonlinearities of the same exponent. However, in the high regularity theory, it is often necessary to impose additional hypotheses on F, for instance that ${\displaystyle \nabla F}$ is a power type nonlinearity of order p-1.

Power type nonlinearities feature in the most general form of the NLS, NLW, and NLKG equations.

Special types of power nonlinearity

A power type nonlinearity ${\displaystyle F:\mathbb {C} \to \mathbb {C} }$ is U(1)-invariant if

${\displaystyle F(e^{i\theta }u)=e^{i\theta }F(u)}$

for all real ${\displaystyle \theta }$ and complex u. In the context of the NLS, this ensures that the equation enjoys both phase rotation invariance and Galilean invariance. Thus the pure power nonlinearities ${\displaystyle F(u)=\pm |u|^{p-1}u}$ are U(1)-invariant, but other polynomials such as ${\displaystyle u^{p}}$ or ${\displaystyle |u|^{p}}$ are not.

A power type nonlinearity ${\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ is conservative or Hamiltonian if there exists a potential ${\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} }$ such that ${\displaystyle F(u)=\nabla V(u)}$. This definition can be complexified by identifying complex spaces with real spaces of twice the dimension. For instance, the pure power nonlinearities ${\displaystyle F(u)=\pm |u|^{p-1}u}$ are Hamiltonian with potential ${\displaystyle V(u)=\pm {\frac {|u|^{p+1}}{p+1}}}$. Normalizing ${\displaystyle V(0)=0}$, we see that the potential V is thus a power type nonlinearity of order ${\displaystyle p+1}$. A Hamiltonian nonlinearity ${\displaystyle F:\mathbb {C} \to \mathbb {C} }$ is U(1)-invariant if and only if its potential function V(u) is radial, i.e. it depends only on the magnitude |u| of u.

We say that a Hamiltonian nonlinearity is coercive if the potential is non-negative modulo lower order terms. Thus for instance, the defocusing nonlinearity ${\displaystyle F(u)=+|u|^{p-1}u}$ is coercive, but the focusing nonlinearity ${\displaystyle F(u)=+|u|^{p-1}u}$ is not.