Completely Integrable PDEs

Evoluitionary equations

Derivative Schrodinger equation (DSE): q_t = i/2 q_{xx} + 1/2 (q |q|^2)_x

Evolutionary Non Linear Schrodinger (ENLS) : i u_t = u_xx + 2 |u|² u done

Extended nonlinear Schrödinger equation with higher-order odd and even terms (Formula (25)) :

i q_t + ½ q_{xx} + |q|^2 q -

- i \alpha_3 [q_{xxx} + 6q_x|q|^2)\] +

+\gamma [q_{xxxx} +6 (q_x)^2 q*+4q|q_x|^2 +8q_{xx} |q|^2 + 2 q*_{xx}q^2+6q|q|^4 ] = 0 done

This equation in some particular cases becomes

• ENLS equation (\alpha_3=0, \gamma=0) done
• Hirota equation (\gamma=0) done
• Lakshmanan-Porsezian-Daniel equation (\alpha_3=0) done

Generalization of NLS equation: a q_t = i q_{xx} + c q_x + 2 i ( a |q|^2 + d(q +q*)) (d + a q) for real a,c,d done

This equation becomes ENLS if a=1, c=d=0,

Harry Dym (HD) :

Faquir-Manna-Neveu: q_{xt} = q - q q_{xx} - ½ (q_x)^2 + a /2 (q_x)^2 q_{xx}

Fermi–Pasta–Ulam (continuous analogue of) : q_{tt} – q_{xx} - a q_x q_{xx} + b q_{xxxx} = 0

Schafer-Wayne: q_{xt} - q - 1/6 (q^3)_{xx} = 0

2+1 equation

Kadomtzev-Petviashvili (KP) : q_{yy} - q_{xt} + 1/4 q_{xxxx} - 3/2 (qq_x)_x = 0 done partially

Novikov-Veselov:

where and the following standard notations of complex analysis are used:

is the real part, and

Systems of PDEs

Davey–Stewartson equation (DSE):