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Following the work of Fuchs in classifying first-order ordinary differential equations, Painlevé studied second-order ordinary differential equation of the form ...

There are six Painlevé transcendents, corresponding to second-order ordinary differential equations whose only movable singularities are ordinary poles and which cannot be ...

The partial differential equation u_(xt)=sinhu, which contains u_(xt) instead of u_(xx)-u_(tt) and sinhu instead to sinu, as in the sine-Gordon equation (Grauel 1985; ...

A function which is not an algebraic function. In other words, a function which "transcends," i.e., cannot be expressed in terms of, algebra. Examples of transcendental ...

The Ablowitz-Ramani-Segur conjecture states that a nonlinear partial differential equation is solvable by the inverse scattering method only if every nonlinear ordinary ...

The linear Boussinesq equation is the partial differential equation u_(tt)-alpha^2u_(xx)=beta^2u_(xxtt) (1) (Whitham 1974, p. 9; Zwillinger 1997, p. 129). The nonlinear ...

The partial differential equation u_t+uu_x=nuu_(xx) (Benton and Platzman 1972; Zwillinger 1995, p. 417; Zwillinger 1997, p. 130). The so-called nonplanar Burgers equation is ...

The partial differential equation 3/4U_y+W_x=0, (1) where W_y+U_t-1/4U_(xxx)+3/2UU_x=0 (2) (Krichever and Novikov 1980; Novikov 1999). Zwillinger (1997, p. 131) and Calogero ...

A partial differential equation which appears in differential geometry and relativistic field theory. Its name is a wordplay on its similar form to the Klein-Gordon equation. ...