Partial Differential Equations  

Hyperbolic > s.a. types of wave equations; klein-gordon equation.
* Properties: They admit an initial-value formulation.
@ General references: Leray 52; Kundt & Newman JMP(68) [and characteristic propagation]; Beyer gq/05-ln [linear and quasi-linear, semigroup methods]; Hattori 13 [II/III].
@ Non-linear: Li 94 [quasi-linear]; Shatah & Sogge in Chruściel 97; Claudel & Newman PRS(98) [quasi-linear, with singularity in time].
@ Characteristic: Frittelli JPA(04)mp, JPA(05)mp/04 [first-order, stability]; Nicolas m.AP/05 [second-order wave equations].
@ Singularities: Geroch JMP(83) [non-singularity theorem]; Witt 95 [with conical points]; Vickers & Wilson gq/01 [hypersurface singularities]; Chen 10; Ghergu & Taliaferro 16.

Elliptic > s.a. Poisson Equation.
* Steady-state equation: The general stationary limit of both the general wave and the diffusion equations, \[ -\nabla\cdot(p\nabla u) + qu = F(x)\;; \] Special cases are the Poisson equation \(\nabla^2u = -f\) (the case with \(p\) = const, \(q\) = 0, and \(f:= F/p\)) and the laplace equation.
@ Numerical methods: Okawa IJMPA(13)-a1308-ln [and constraints in numerical relativity].
@ Non-linear: in Nirenberg 74; Kiessling PhyA(00)mp [Poisson-Boltzmann, Paneitz]; López-Fontán et al PhyA(07) [Poisson-Boltzmann].

@ References: Friedman 64; Lieberman 96.

Integrability Conditions for a System of PDEs
* Idea: Conditions to be satisfied by the (known) functions appearing in a system of coupled pde's, in order for it to admit solutions.
* Example: If \(u_{,x} = F(x, y)\) and \(u_{,y} = G(x, y)\), need \(F_{,y} = G_{,x}\).

Types and Solution Methods > s.a. Boundary-Value Problems; green functions; Inverse Scattering; laplace equation; symmetries.
* History: The symmetry reduction method of finding group-invariant solutions was proposed by S Lie in the XIX century.
* Symmetries and reduction: Any symmetry reduces a second-order differential equation to a first-order equation; Variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, symmetries of only the equations of motion (exemplified by scale-invariant hydrostatics) yield first-order non-conservation laws between invariants.
* Separation of variables: Leads to ordinary differential equations in eigenvalue form.
@ Superposition of solutions: Zhdanov JPA(94) [non-linear separation of variables]; Cariñena & Ramos AAM-mp/01; > s.a. Ermakov System.
@ Symmetries and reduction: Baumann et al JPA(94) [non-classical]; Anderson et al CMP(00)mp/99, CMP(00)mp/99, mp/01-proc [generalization to non-transverse actions]; Nucci TMP(05) [Lie group analysis]; Gaeta & Mancinelli IJGMP(05)mp/06 [asymptotic symmetries]; Cicogna & Laino RVMP(06)mp [conditional symmetries]; Bludman a1106 [non-variational symmetries]; Steinberg & de Melo Marinho a1409-ln [computational approach].
@ Non-symmetric solutions: Martina et al JPA(01)mp [infinite-dimensional symmetry group].
@ First-order: Holcman & Kupka QJM(05)mp/03 [on compact manifolds]; Bogoyavlenskij CMP(96) [existence of Hamiltonian structures]; López 12.
@ Second-order: LaChapelle AP(04)mp, AP(04)mp [linear, path-integral method]; Cioranescu et al 18.
@ Linear: Hörmander 85; Polyanin 01 [handbook].
@ Non-linear: Adomian 94; Kong & Hu PLA(98) [solutions, geometric]; Ramm MMMAA-mp/00; Ludu et al IJCMS-mp/02 [multiscale analysis]; Peng PLA(03) [including sine-Gordon]; Fairlie JPA(04)mp, JNMP(05)mp/04 [implicit solutions]; Lü PLA(06) [Burgers equation-based solutions]; Khater et al IJTP(06) [conservation laws]; Torres-Córdoba a0709 [Monge equation, solution]; Sals & Gómez a0805 [coupled systems]; Polyanin & Zaitsev 11 [handbook]; Anco et al Sigma(11)-a1105 [method of group foliation reduction]; Maheswari & Sahadevan JPA(11) [conservation laws]; Debnath 12; Tadmor BAMS(12) [numerical methods]; Li & Song 16 [variational methods]; > s.a. Riemann Equation; types of wave equations.
@ Stochastic: Hochberg et al PRE(99)cm [stochastic noise]; Hairer Nonlin(02)-mp/01 [reaction-diffusion]; > s.a. effective potential; stochastic processes.
@ Other types: Visser & Yunes IJMPA(03)gq/02 [scale-invariant]; Barnaby & Kamran JHEP(08)-a0709, JHEP(08)-a0809 [infinitely many derivatives, initial-value problem]; Tehseen & Prince JPA(13)-a1302 [using differential geometric methods]; Ablinger et al a1608 [coupled systems, in terms of power series]; > s.a. differential equations [fractional], Combinatorial PDEs.
@ Spectral methods: Bonazzola et al JCAM(99)gq/98 [in general relativity]; Piotrowska et al a1712 [non-smooth problems].

In Mathematical Physics > s.a. chaotic systems; diffusion.
@ General references: Rubinstein 94; Geroch gq/96; Calin & Chang 04 [on Riemannian manifolds]; Kirkwood 12; Lein a1508-ln.
@ Geometrical aspects: Zharinov 92; Gràcia et al IJGMP(04)mp.
@ Hyperbolicity: Gundlach & Martín-García PRD(04)gq [symmetric]; Reula gq/04 [strong]; Beig LNP(06)gq/04 [rev].
> Gravity-related examples: see einstein's equation; initial-value formulation of general relativity; numerical relativity.

Other References > s.a. differential equations.
@ General: Webster 47; Sommerfeld 49; Ayres 52; Petrovsky 54 (reprint 91); Bers et al 64; Garabedian 64; Meis & Marcowitz 81; John 82; Bellman & Adomian 85; Zachmanoglou & Thoe 86; Stephani 89; Hubbard & West 90; Cronin 94; Xu a1205 [algebraic approaches].
@ Books, III: Edelen & Wang 92; Folland 95; Stephenson 96; Christodoulou 00.
@ And Lie groups: Olver 93; Dresner 98.
@ Initial-value formulation: Bers et al 64 [non-second-order diagonal].
@ Conservation laws: Anco & Bluman EJAM(02)mp/01, EJAM(02)mp/01; Zhang a1409 [new technique]; Anco & Kara EJAM-a1510 [symmetry invariance].
@ Computational: Wolf EJAM(02)cs.SC/03 [conservation laws]; Hawley & Matzner CQG(04)gq/03 [elliptic equations and holes]; Valiquette & Winternitz JPA(05)mp [discretization and symmetries]; Formaggia et al 12; Bartels 15 [non-linear]; > s.a. computational physics; Finite-Element Method; numerical relativity; Courant-Friedrichs-Lewy Condition.
@ With Mathematica: Vvedensky 92; Ross 04; Stavroulakis & Tersian 04.
@ Handbook: Zwillinger 89; Polyanin et al 01 [first-order].
@ Related topics: Werschulz 91 [complexity]; Medvedev PRS(99) [Poincaré normal form]; Evans BAMS(04) [entropy methods].

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