Null structure

A semilinear equation is said to have null structure if the resonant component of the nonlinearity vanishes; in other words, plane waves obeying the dispersion relation cannot interact via the nonlinearity to generate forcing terms which also obey the dispersion relation. This generally requires the nonlinearity to be a linear combination of a certain special set of nonlinear forms, known as null forms. The term is primarily used for non-linear wave equations, but also applies to a number of other nonlinear dispersive equations.

Quasilinear equations can also exhibit null structure, although the precise definition of this concept in this case is still not fully understood. For instance, the Einstein equations do not exhibit classical null structure, because the (resonant) self-interaction of plane waves is non-trivial, but nevertheless enjoys a kind of "nilpotent" null structure which can achieve a similar effect as classical null structure.

Null forms in wave equations

The non-linear expressions which occur in non-linear wave equations often have null form structure. Roughly speaking, this means that travelling waves $exp(i(k.x+-|k|t))$ do not self-interact, or only self-interact very weakly. When one has a null form present, the local and global well-posedness theory often improves substantially. There are several reasons for this. One is that null forms behave better under conformal compactification. Another is that null forms often have a nice representation in terms of conformal Killing vector fields. Finally, bilinear null forms enjoy much better estimates than other bilinear forms, as the interactions of parallel frequencies (which would normally be the worst case) is now zero.

The standard bilinear null forms are

$Q_{0}(\phi ,\psi ):=\partial ^{\alpha }\phi \partial _{\alpha }\psi =-\phi _{t}\psi _{t}+\nabla \phi \cdot \nabla \psi$ $Q_{0i}(\phi ,\psi ):=\phi _{t}\psi _{i}-\phi _{i}\psi _{t}$ $Q_{ij}(\phi ,\psi ):=\phi _{i}\psi _{j}-\phi _{j}\psi _{i}.$

The only bilinear forms in first derivatives of scalar solutions to the wave equation which have null structure are linear combinations of the above forms. However, more null forms are possible if one has trilinear or higher nonlinearities, if higher derivatives of solutions are permitted, or if one is considering vector or tensor equations (such as Maxwell or Dirac type equations) which obey additional compatibility conditions.

The presence of null structure seems to be related to the covariance of the underlying equation or Lagrangian, although the exact connection is not well understood.

Equations with null structure

• Dirac-Klein-Gordon equation
• Einstein equation
• Maxwell-Dirac equation
• Maxwell-Klein-Gordon system
• Korteweg-de Vries equation (and more general KdV-type equations)
• nonlinear Dirac equation
• Schrodinger maps
• Wave maps
• Yang-Mills equations