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- The '''Hartree equation''' is of the form ...c Schrodinger equation; the [[cubic NLS]] is in some sense a limit of this equation as <math>n \rightarrow d,</math> (after suitable normalization of the kerne3 KB (469 words) - 17:30, 9 July 2007
- #REDIRECT [[Semilinear Schrodinger equation]]45 bytes (4 words) - 03:05, 29 July 2006
- 29 bytes (3 words) - 06:34, 27 July 2006
- 32 bytes (3 words) - 00:45, 10 August 2006
- 31 bytes (3 words) - 07:09, 31 July 2006
- The (homogeneous) '''Airy equation''' is given by ...tions. One such problem is the reduced (or linearized) [[Korteweg-de Vries equation]]1 KB (214 words) - 22:11, 19 January 2018
- The theory of the '''Schrodinger equation with potential'''285 bytes (43 words) - 04:51, 4 August 2006
- ...equation]], and when <math>\lambda > 0</math> the equation is the [[KP-II equation]]. These equations are a model for shallow long waves in the x direction, ...varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least). KP-I also arises as a model for sound waves in ferro3 KB (527 words) - 19:46, 3 February 2011
- 23 bytes (3 words) - 06:38, 29 July 2006
- 34 bytes (3 words) - 15:45, 5 May 2007
- The '''free Schrodinger equation''' ...dimensions. One can also consider the inhomogeneous (forced) Schrodinger equation2 KB (307 words) - 07:45, 15 June 2009
- 38 bytes (4 words) - 17:01, 30 July 2006
- The '''free wave equation''' on <math>{\mathbb R}^{1+d}</math> is given by One can add a mass term to create the [[Klein-Gordon equation]].2 KB (358 words) - 16:00, 1 July 2018
- The '''Klein-Gordon equation''' is given by [[Minkowski space]]. When the mass is zero this is just the [[free wave equation]]. When the mass is positive, one usually rescales spacetime to normalize1 KB (169 words) - 20:56, 10 June 2007
- 29 bytes (3 words) - 22:12, 3 September 2007
- 29 bytes (3 words) - 22:12, 3 September 2007
- The (hyperbolic) '''minimal surface equation''' takes the form ...n \times R</math> ). This is the Minkowski analogue of the minimal surface equation in Euclidean space, see [[Hp1994]].809 bytes (125 words) - 22:10, 10 April 2007
- ...ints of that functional. It can be computed formally by starting with the equation ...iltonian]] can (in principle, at least) be expressed as the Euler-Lagrange equation of a functional, and conversely Euler-Lagrange equations can in principle b657 bytes (95 words) - 20:14, 11 June 2007
- The '''sine-Gordon equation''' ...ontinuum of pendula hanging from a wire. It is a [[completely integrable]] equation, and has many interesting solutions, including "breather" solutions.705 bytes (103 words) - 23:39, 22 January 2009
- ...nt the electromagnetic field. x and v live in R^3 and t lives in R. This equation is a coupled wave and conservation law system, and models collision-less pl ...\rho</math>. Considerably more is known for the existence theory of this equation.2 KB (297 words) - 13:33, 11 July 2007
- The '''KP-I equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the parameter <math>\lambda</math> is negative. Well-posedness is u *If one considers the fifth-order KP-I equation (replace <math>u_{xxx}</math> by <math>u_{xxxxx}</math>) then one has GWP i3 KB (537 words) - 06:21, 21 July 2007
- #REDIRECT [[Kadomtsev-Petviashvili equation]]45 bytes (3 words) - 04:59, 27 July 2006
- The '''KP-II equation''' is the special case of the [[Kadomtsev-Petviashvili equation]] when the The KP-II equation can be generalized to three dimensions (replace partial_yy with partial_yy2 KB (264 words) - 01:22, 17 March 2007
- <center>'''Benjamin-Ono equation'''</center> The Benjamin-Ono equation (BO) [[Bj1967]], [[On1975]], which models one-dimensional internal waves in4 KB (707 words) - 18:49, 25 October 2008
- {{equation | equation = <math>u_t + u_{xxx} = \pm u^3 u_x</math>2 KB (376 words) - 22:26, 4 March 2007
- {{equation | equation = <math>u_t + u_{xxx} = \pm u^4 u_x</math>3 KB (496 words) - 22:03, 4 March 2007
- {{equation | equation = <math>iu_t + \Delta u = \pm |u|^{p-1} u</math>2 KB (349 words) - 05:44, 21 July 2007
- #REDIRECT [[Hartree equation]]30 bytes (3 words) - 04:29, 29 July 2006
- 29 bytes (3 words) - 22:12, 3 September 2007
- The '''Korteweg-de Vries (KdV) equation''' is The equation is [[completely integrable]], and has infinitely many conserved quantities.2 KB (307 words) - 16:10, 14 September 2006
- #REDIRECT [[Ginzburg-Landau-Schrodinger equation]]50 bytes (3 words) - 06:36, 16 January 2007
- 44 bytes (3 words) - 15:47, 5 May 2007
- 34 bytes (5 words) - 06:35, 27 July 2006
- The '''Ginzburg-Landau-Schrodinger''' equation is The main focus of study for this equation is the formation of vortices and their dynamics in the limit <math>\epsilon2 KB (368 words) - 07:45, 2 October 2007
- #REDIRECT [[Ginsberg-Landau-Schrodinger equation]]50 bytes (3 words) - 14:24, 9 August 2006
- 34 bytes (5 words) - 06:34, 27 July 2006
- The (defocusing) '''modified Korteweg-de Vries (mKdV) equation''' is ...roughly equivalent to the <span class="SpellE">H^k</span> norm of u. This equation has been studied [[modified Korteweg-de Vries on R|on the line]], [[modifie987 bytes (149 words) - 07:41, 31 July 2006
- ...particularly important class of such equations is the [[Schrodinger maps]] equation. ...ates which can recover a full derivative for the inhomogeneous Schrodinger equation providing there is sufficient decay in space, and so one can still obtain w5 KB (844 words) - 10:50, 24 August 2006
- 56 bytes (5 words) - 23:09, 14 August 2006
- 36 bytes (5 words) - 06:35, 27 July 2006
- 36 bytes (5 words) - 06:35, 27 July 2006
- 5 KB (777 words) - 20:49, 10 June 2007
- The local and global [[well-posedness]] theory for the [[Korteweg-de Vries equation]] on the real line is as follows. The <span class="SpellE">KdV</span> equation can also be generalized to a 2x2 system4 KB (617 words) - 21:51, 11 September 2007
- The local and global [[well-posedness]] theory for the [[Korteweg-de Vries equation]] on the torus is as follows. ...(at least in principle) from work on the <span class="SpellE">mKdV</span> equation by [Takaoka and Tsutsumi?]2 KB (386 words) - 22:51, 14 August 2006
- The local and global [[well-posedness]] theory for the [[Korteweg-de Vries equation]] on the half-line is as follows.2 KB (369 words) - 14:30, 10 August 2006
- '''Liouville's equation''' It is a limiting case of the [[Sine-Gordon|sinh-gordon equation]].2 KB (294 words) - 10:16, 27 January 2009
Page text matches
- A '''fully nonlinear''' equation takes the form for some nonlinear function ''F''. By differentiating this equation once and working with the system <math>v := (u, Du)</math>386 bytes (62 words) - 00:46, 8 August 2006
- * The [[Ginsberg-Landau-Schrodinger equation]] * The [[Benjamin-Ono equation]]832 bytes (85 words) - 14:27, 9 August 2006
- ...e vanishing mass limit of the [[Klein-Gordon equation]] is the [[free wave equation]].261 bytes (40 words) - 04:52, 4 August 2006
- ...proximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation262 bytes (28 words) - 20:59, 17 June 2009
- ...valence of a nonlinear Schrödinger equation and a Heisenberg ferromagnetic equation267 bytes (32 words) - 00:13, 11 July 2009
- ...dified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation297 bytes (36 words) - 00:10, 24 May 2009
- ...near estimates related with the KdV equation and the nonlinear Schrödinger equation317 bytes (37 words) - 18:01, 10 June 2009
- The '''Klein-Gordon equation''' is given by [[Minkowski space]]. When the mass is zero this is just the [[free wave equation]]. When the mass is positive, one usually rescales spacetime to normalize1 KB (169 words) - 20:56, 10 June 2007
- ...tion on the non-periodic domain <math>\mathbf{R}^d</math>, though when the equation has [[infinite speed of propagation]] then the justification of this limit430 bytes (70 words) - 04:56, 4 August 2006
- ...ely integrable in the sense that there exist a Lax pair formulation of the equation. a completely integrable equation. Some nonlinear PDE's as the KdV posed1 KB (166 words) - 08:40, 4 February 2011
- The '''sine-Gordon equation''' ...ontinuum of pendula hanging from a wire. It is a [[completely integrable]] equation, and has many interesting solutions, including "breather" solutions.705 bytes (103 words) - 23:39, 22 January 2009
- ...equation on the line and inverse scattering for the nonlinear Schrodinger equation with a potential319 bytes (42 words) - 20:26, 6 July 2009
- ...operation which converts solutions of one equation to solutions of another equation. Usually one only works with transforms which are given explicitly by an a302 bytes (43 words) - 00:13, 15 August 2006
- ...ints of that functional. It can be computed formally by starting with the equation ...iltonian]] can (in principle, at least) be expressed as the Euler-Lagrange equation of a functional, and conversely Euler-Lagrange equations can in principle b657 bytes (95 words) - 20:14, 11 June 2007
- ...the presence of non-compact symmetries (or approximate symmetries) in the equation and in the functional, however this can sometimes be counteracted using [[c899 bytes (129 words) - 01:41, 15 August 2006
- The (hyperbolic) '''minimal surface equation''' takes the form ...n \times R</math> ). This is the Minkowski analogue of the minimal surface equation in Euclidean space, see [[Hp1994]].809 bytes (125 words) - 22:10, 10 April 2007
- ...lize the solutions to have frequency comparable to one. In that case, the equation acquires a rescaling parameter <math>\hbar</math>, and the limit is usually867 bytes (126 words) - 20:02, 5 August 2006
- ...forward in time. The model instance of a dissipative equation is the heat equation ...ispersive equations are often [[Hamiltonian]] and are the [[Euler-Lagrange equation]] for some [[variational problem]], dissipative equations are usually non-H869 bytes (123 words) - 01:19, 15 August 2006
- The (homogeneous) '''Airy equation''' is given by ...tions. One such problem is the reduced (or linearized) [[Korteweg-de Vries equation]]1 KB (214 words) - 22:11, 19 January 2018
- ...asymptotically as <math>t \to \pm \infty</math> to solutions of the linear equation. Scattering can be formalized by rigorous results establishing the existen394 bytes (54 words) - 21:11, 9 August 2006
- The (defocusing) '''modified Korteweg-de Vries (mKdV) equation''' is ...roughly equivalent to the <span class="SpellE">H^k</span> norm of u. This equation has been studied [[modified Korteweg-de Vries on R|on the line]], [[modifie987 bytes (149 words) - 07:41, 31 July 2006
- #REDIRECT [[Hartree equation]]30 bytes (3 words) - 04:29, 29 July 2006
- '''Liouville's equation''' It is a limiting case of the [[Sine-Gordon|sinh-gordon equation]].2 KB (294 words) - 10:16, 27 January 2009
- #REDIRECT [[Kadomtsev-Petviashvili equation]]45 bytes (3 words) - 04:59, 27 July 2006
- #REDIRECT [[Semilinear Schrodinger equation]]45 bytes (4 words) - 03:05, 29 July 2006
- #REDIRECT [[Semilinear Schrodinger equation]]45 bytes (4 words) - 04:02, 29 July 2006
- #REDIRECT [[GKdV-3 equation]]29 bytes (3 words) - 20:06, 9 August 2006
- #REDIRECT [[GKdV-4 equation]]29 bytes (3 words) - 20:06, 9 August 2006
- #REDIRECT [[Benjamin-Ono equation]]35 bytes (3 words) - 18:38, 10 August 2006
- #REDIRECT [[Sine-Gordon equation]]34 bytes (3 words) - 15:50, 5 May 2007
- The '''free Schrodinger equation''' ...dimensions. One can also consider the inhomogeneous (forced) Schrodinger equation2 KB (307 words) - 07:45, 15 June 2009
- A '''quasilinear equation''' is an equation of the form ...nontrivial) in the top order terms <math>D^k u</math>. Thus a quasilinear equation takes the schematic form907 bytes (155 words) - 15:35, 12 December 2007
- #REDIRECT [[Korteweg-de Vries equation]]40 bytes (4 words) - 06:24, 31 July 2006
- #REDIRECT [[Ginsberg-Landau-Schrodinger equation]]50 bytes (3 words) - 14:24, 9 August 2006
- #REDIRECT [[Ginzburg-Landau-Schrodinger equation]]50 bytes (3 words) - 06:36, 16 January 2007
- #REDIRECT [[Generalized Korteweg-de Vries equation]]52 bytes (5 words) - 06:14, 31 July 2006
- #REDIRECT [[Derivative non-linear Schrodinger equation]]56 bytes (5 words) - 07:27, 31 July 2006
- #REDIRECT [[modified Korteweg-de Vries equation]]49 bytes (5 words) - 07:40, 31 July 2006
- #REDIRECT [[Generalized Korteweg-de Vries equation]]52 bytes (5 words) - 20:05, 9 August 2006
- #REDIRECT [[Korteweg-de Vries equation on T]]45 bytes (6 words) - 05:00, 8 August 2006
- #REDIRECT [[Korteweg-de Vries equation on R]]45 bytes (6 words) - 05:02, 8 August 2006
- ...larization method''' or '''viscosity method''') constructs solutions to an equation by the following general procedure: # Regularize the equation somehow (for instance, by adding penalty terms to the Hamiltonian or Lagran1 KB (174 words) - 01:03, 15 August 2006
- B. Birnir, G. Ponce, L. Vega, ''The local ill-posedness of the modified KdV equation. Annales de l'I. H. P.'', '''13''' (1996), 529-535. [http://archive.numdam. ...posedness for the initial value problem for the modified Korteweg-de Vries equation'', preprint.462 bytes (63 words) - 22:52, 26 February 2010
- #REDIRECT [[Korteweg-de Vries equation on the half-line]]57 bytes (7 words) - 20:03, 9 August 2006
- ...linear wave equation]] and a [[Schrodinger equations|nonlinear Schrodinger equation]]. The main examples are: ** The [[magnetic Zakharov equation]] (formed by adding a magnetic field to the Zakharov system)748 bytes (101 words) - 20:24, 9 August 2006
- ...nal equation; perturbation theory (and additional smoothing effects of the equation) is then used to convert the approximate solution to an exact one. The per ...ightly different method of mollifying the high frequency components of the equation and analyzing the energy of the resulting field, rather than1 KB (182 words) - 06:40, 16 January 2007
- The '''Korteweg-de Vries (KdV) equation''' is The equation is [[completely integrable]], and has infinitely many conserved quantities.2 KB (307 words) - 16:10, 14 September 2006
- When the linear equation is constant-coefficient then the Fourier transform is an excellent tool for ...be extracted by considering the [[small amplitude limit]] of the original equation.1 KB (145 words) - 04:29, 8 August 2006
- ...ve or a solution independent of the spatial variables) which collapses the equation to an ODE. This is a special case of the method of separation of variables ...p solutions to various equations such as the focusing [[NLW|nonlinear wave equation]].450 bytes (72 words) - 05:02, 4 August 2006
- H. Smith, Small data scattering for a nonlinearly damped wave equation, preprint.109 bytes (14 words) - 17:40, 8 August 2006
- ...al, F. Linares, ''Some properties for a higher order nonlinear Schrodinger equation'', Preprint.132 bytes (16 words) - 01:35, 1 August 2006
- ....org/abs/math.AP/0311227 Instability of the periodic nonlinear Schrodinger equation]'', preprint.173 bytes (24 words) - 01:48, 1 August 2006
- ...f which are perturbations of one sort or another of the [[free Schrodinger equation]]. One general class of such equations takes the form ...n derivatives then we refer to this equation as a [[semilinear Schrodinger equation]] (NLS). These equations (particularly the [[cubic NLS]]) arise as model eq4 KB (615 words) - 03:44, 8 February 2011
- H. Smith, D. Tataru, Counterexamples to Strichartz estimates for the wave equation with nonsmooth coefficients, preprint.149 bytes (18 words) - 17:37, 8 August 2006
- B. Yordanov, Blow-up for the one-dimensional Klein-Gordon equation with a cubic non-linearity, preprint.132 bytes (15 words) - 15:55, 8 August 2006
- I. Rodnianski, T. Tao, Global regularity for the Maxwell-Klein-Gordon equation in high dimensions, preprint.136 bytes (16 words) - 17:59, 8 August 2006
- ...Vries on R|complex mKdV]], and a cubic nonlinear [[Airy equation]]. This equation is a general model for propagation of pulses in an optical fiber [[Kod1985]1 KB (197 words) - 12:45, 11 July 2007
- ...[[Schrodinger maps|Schrodinger maps]], [[Cubic DNLS on R|DNLS]], [[Hartree equation|Hartree]], etc.) ...odified Korteweg-de Vries equation|mKdV]], [[Generalized Korteweg-de Vries equation|gKdV]], etc.)2 KB (321 words) - 18:34, 10 June 2018
- ...may be only a distribution) is then the '''fundamental solution''' of the equation. Another way of viewing this is that the fundamental solution is the solut589 bytes (90 words) - 04:32, 10 August 2006
- C.D. Sogge, Estimates for the Dirichlet-wave equation and applications to nonlinear wave equations, preprint.137 bytes (17 words) - 17:35, 8 August 2006
- The '''low-frequency limit''' of an equation concerns the behaviour of the equation for solutions which have very low frequency <math>|\xi| \to 0</math>.672 bytes (99 words) - 20:05, 5 August 2006
- ...foschi/t.pdf On the regularity of multilinear forms associated to the wave equation]'', PhD Thesis, January 2000.180 bytes (28 words) - 02:45, 1 August 2006
- ...estimates in hyperbolic space and global existence for the semilinear wave equation, TAMS 353 (2001), 785-807.160 bytes (19 words) - 16:22, 8 August 2006
- ...B. Zhang, X. Zhang, ''A note on self-similar solutions of the Schrodinger equation'', preprint.188 bytes (26 words) - 17:48, 8 August 2006
- Solutions to the [[Airy equation]] and [[KdV equations|its perturbations]] are either estimated in mixed spa ...nger equation]] in [[Bo1993b]], although the analogous spaces for the wave equation appeared earlier [[RaRe1982]], [[Be1983]] in the context of [[propagation o1 KB (189 words) - 19:58, 14 August 2006
- | title = On the Korteweg-de Vries equation192 bytes (20 words) - 23:04, 26 May 2009
- | title = Periodic nonlinear Schrödinger equation and invariant measures213 bytes (22 words) - 20:53, 7 May 2009
- | title = Inverse scattering for the nonlinear Schrödinger equation209 bytes (22 words) - 20:24, 6 July 2009
- | title = Solutions of the wave equation with localized energy204 bytes (22 words) - 23:46, 15 June 2009
- C. Sulem, P. Sulem, The nonlinear Schrodinger equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences 139, Spring178 bytes (20 words) - 16:36, 8 August 2006
- A global or integral '''conservation law''' for an evolution equation is any quantity ''Q(t)'' depending on the value of all the fields at time ' ...quantities to symmetries of the underlying equation, in the case that the equation is [[Hamiltonian]] or Lagrangian.2 KB (275 words) - 18:47, 15 September 2006
- | title = On well-posedness of the forced nonlinear Schrödinger equation200 bytes (23 words) - 02:11, 10 May 2009
- | title = Blow up and scattering in the nonlinear Schrödinger equation212 bytes (24 words) - 00:22, 11 May 2009
- The '''small amplitude limit''' for a nonlinear equation arises when considering initial position ...s, the small amplitude limit is usually just the linear counterpart of the equation; however when analyzing long times (e.g. times comparable to <math>1/\epsil724 bytes (111 words) - 05:23, 2 August 2006
- | title = Periodic nonlinear Schrödinger equation and invariant measures228 bytes (24 words) - 20:58, 7 May 2009
- | title = Decay estimate for the wave equation with potential218 bytes (24 words) - 19:36, 17 May 2009
- | title = On the derivative nonlinear Schrödinger equation212 bytes (24 words) - 00:21, 24 May 2009
- | title = Global solvability of the derivative nonlinear Schrodinger equation219 bytes (23 words) - 21:19, 5 June 2009
- | title = Unique continuation for the KdV equation201 bytes (23 words) - 21:10, 9 July 2009
- | title = Well-posedness for the Kadomtsev-Petviashvili II equation219 bytes (22 words) - 01:13, 30 June 2009
- | title = Regularity of solutions to the Schrödinger equation212 bytes (25 words) - 00:39, 21 June 2009
- | title = Scattering Theory for Nonlinear Klein-Gordon Equation with Sobolev Critical Power232 bytes (24 words) - 17:52, 10 June 2009
- ...n ODE of the form <math>-c f + {\frac{d^2}{dx^2}} f + f^2 = 0</math>. This equation can be explicitly solved in terms of the hyperbolic secant function. Other ...uations, and it is suspected (but unproven for any focusing non-integrable equation) that solutions for such equations, if they do not blow up, should generica2 KB (236 words) - 14:24, 24 August 2009
- | title = An example of blowup at infinity for a quasilinear wave equation228 bytes (26 words) - 00:24, 11 July 2009
- | title = Semiclassical limit of the nonlinear Schrödinger equation in small time225 bytes (25 words) - 00:54, 22 May 2009
- | title = On the generalized Benjamin-Ono equation211 bytes (23 words) - 18:26, 30 May 2009
- | title = On a nonlinear dispersive equation with time-dependent coefficients225 bytes (24 words) - 21:02, 5 June 2009
- | title = Global low-regularity solutions for Kadomtsev-Petviashvili equation.231 bytes (22 words) - 18:02, 4 July 2009
- | title = On global existence and scattering for the wave maps equation210 bytes (24 words) - 17:54, 4 July 2009
- | title = The Korteweg-de Vries equation: A survey of results207 bytes (23 words) - 22:42, 8 June 2009
- | title = Uniqueness of weak solution of the KdV equation195 bytes (22 words) - 21:16, 9 July 2009
- | title = Invariant measures for the 2d-defocussing nonlinear Schrödinger equation239 bytes (25 words) - 21:03, 7 May 2009
- | title = Periodic Korteweg de Vries equation with measures as initial data233 bytes (27 words) - 23:52, 18 January 2018
- ...Resonance and long time existence for the quadratic nonlinear Schrödinger equation231 bytes (26 words) - 01:07, 13 May 2009
- | title = On the initial value problem for a nonlinear Schrödinger equation223 bytes (29 words) - 01:34, 15 May 2009
- | title = Profile decomposition for the wave equation outside a convex obstacle230 bytes (26 words) - 00:49, 17 May 2009
- | title = Global solution to the two-dimensional Klein-Gordon equation225 bytes (23 words) - 19:35, 17 May 2009
- | title = Regularity for the wave equation with a critical nonlinearity, Comm232 bytes (26 words) - 00:17, 22 May 2009
- | title = Cauchy problem for a generalized nonlinear dispersive equation218 bytes (24 words) - 00:50, 23 May 2009
- | title = Scattering and self-similar solutions for the nonlinear wave equation251 bytes (25 words) - 23:07, 24 May 2009
