819:. A slightly more complicated case is the self dual Yang–Mills equations, when the moduli space is finite-dimensional but not necessarily compact, though it can often be compactified explicitly. Another case when one can sometimes hope to describe all solutions is the case of completely integrable models, when solutions are sometimes a sort of superposition of
768:
The basic questions about singularities (their formation, propagation, and removal, and regularity of solutions) are the same as for linear PDE, but as usual much harder to study. In the linear case one can just use spaces of distributions, but nonlinear PDEs are not usually defined on arbitrary
814:
for example, as this would involve describing all possible fluid motions.) If the equation has a very large symmetry group, then one is usually only interested in the moduli space of solutions modulo the symmetry group, and this is sometimes a finite-dimensional compact manifold, possibly with
835:
It is often possible to write down some special solutions explicitly in terms of elementary functions (though it is rarely possible to describe all solutions like this). One way of finding such explicit solutions is to reduce the equations to equations of lower dimension, preferably ordinary
747:
A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a
857:
Numerical solution on a computer is almost the only method that can be used for getting information about arbitrary systems of PDEs. There has been a lot of work done, but a lot of work still remains on solving certain systems numerically, especially for the Navier–Stokes and
809:
Ideally one would like to describe the (moduli) space of all solutions explicitly, and for some very special PDEs this is possible. (In general this is a hopeless problem: it is unlikely that there is any useful description of all solutions of the
1000:) are invariant under diffeomorphisms of the underlying manifold. Any such symmetry groups can usually be used to help study the equations; in particular if one solution is known one can trivially generate more by acting with the symmetry group.
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The solutions in a neighborhood of a known solution can sometimes be studied by linearizing the PDE around the solution. This corresponds to studying the tangent space of a point of the moduli space of all solutions.
934:
727:. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem.
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equation is not quite parabolic, but is "parabolic modulo the action of the diffeomorphism group", which implies that it has most of the good properties of parabolic equations.
719:. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the
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PDEs that arise from integrable systems are often the easiest to study, and can sometimes be completely solved. A well-known example is the
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for a variational problem. Systems of this form can sometimes be solved by finding an extremum of the original variational problem.
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Spectral transform and solitons. Vol. I. Tools to solve and investigate nonlinear evolution equations
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Sometimes equations are parabolic or hyperbolic "modulo the action of some group": for example, the
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showed that while short time solutions exist, singularities will usually form after a finite time.
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then it usually has an infinite number of first integrals, which help to study it.
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Methods for studying nonlinear partial differential equations
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singularities; for example, this happens in the case of the
1147:, Boca Raton, FL: Chapman & Hall/CRC, pp. xx+814,
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Nonlinear
Partial Differential Equations with Applications
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Some equations have several different exact solutions.
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Handbook of nonlinear partial differential equations
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An example of singularity formation is given by the
756:to the Navier–Stokes equations is one of the seven
1231:(3rd ed.), Boston, MA: Academic Press, Inc.,
1143:Polyanin, Andrei D.; Zaitsev, Valentin F. (2004),
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1072:. New York: John Wiley & Sons. pp. 8–11.
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1017:List of nonlinear partial differential equations
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1132:Encyclopedia of Mathematics
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1179:10.1007/978-3-0348-0513-1
805:Moduli space of solutions
758:Millennium Prize problems
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998:Einstein field equations
949:Euler–Lagrange equations
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817:Seiberg–Witten equations
341:Picard–Lindelöf theorem
335:Existence and uniqueness
1307:Exactly solvable models
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567:Variation of parameters
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346:Peano existence theorem
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752:. The open problem of
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448:Dirac delta function
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788:'s solution of the
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