Power type
A nonlinear function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F: \R^n \to \R^m} is of power type with exponent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p > 1} if one has the bounds
(so in particular Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(0)=\nabla F(0)=0} ) and
for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u, v \in \R^n} and some constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C > 0} . Note that the first bound is a special case of the second once one assumes that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(0)=0} . If F is continuously differentiable, then the second bound is equivalent to the bound
(possibly for a slightly different value of C), thanks to the fundamental theorem of calculus identity
The model example of a power type nonlinearity is the pure power nonlinearity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F: \mathbb{C} \to \mathbb{C}} is U(1)-invariant if
for all real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(u) = \pm |u|^{p-1} u} are U(1)-invariant, but other polynomials such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u^p} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u|^p} are not.
A power type nonlinearity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F: \R^n \to \R^n} is conservative or Hamiltonian if there exists a potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V: \R^n \to \R} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(u) = \pm |u|^{p-1} u} are Hamiltonian with potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V(u) = \pm \frac{|u|^{p+1}}{p+1}} . Normalizing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V(0)=0} , we see that the potential V is thus a power type nonlinearity of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p+1} . A Hamiltonian nonlinearity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(u) = + |u|^{p-1} u} is coercive, but the focusing nonlinearity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(u) = + |u|^{p-1} u} is not.
