Evoluitionary equations
Derivative Schrodinger equation (DSE): q_t = i/2 q_{xx} + 1/2 (q |q|^2)_x
Evolutionary Non Linear Schrodinger (ENLS) : i ut = uxx + 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
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,
Korteweg-de Vries (KdV): ut -¼ uxx + 3/2 u ux = 0 done
1+1 equations
Boussinesq: 3 q_{tt} - q_{xxxx} + 4 (q^2)_{xx} = 0 done
Canonical PDE : b_{tt} = [ - 1/2 (b_x)^2 - ¼ b_{xxx} + (b_t^2 + ¼ b_{xx}^2)/b_x ]_x. done (Obtained from the evolution of canonical systems).
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
where and the following standard notations of complex analysis are used:
is the real part, and
Systems of PDEs
Davey–Stewartson equation (DSE):