Camassa–Holm equation

Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and are intensively studied in the recent literature (see, f.i., [1]-[5]).

Examples

Dispersionless KP equation

The dispersionless Kadomtsev–Petviashvili equation (dKPE) has the form

$(u_t+u{u}_x{)}_x+u_{yy}=0,(1)$

It arises from the commutation

$[L_1,L_2]=0.(2)$

of the following pair of 1-parameter families of vector fields

$L_1={\partial }_y+\lambda {\partial }_x-u_x{\partial }_{\lambda },(3a)$

$L_2={\partial }_t+({\lambda }^2+u){\partial }_x+(-\lambda u_x+u_y){\partial }_{\lambda },(3b)$

where $\lambda$ is a spectral parameter. The dKPE is the $x$-dispersionless limit of the celebrated Kadomtsev–Petviashvili equation.

Dispersionless Korteweg–de Vries equation

The dispersionless Korteweg–de Vries equation (dKdVE) reads as

$u_t=\frac{3}{2}u{u}_x.(4)$

It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation.

Dispersionless Novikov–Veselov equation

The dispersionless Novikov-Veselov equation is most commonly written as the following equation on function $v=v(x_1,x_2,t)$:

$\begin{array}{rl}& {\partial }_{t}v={\partial }_z(vw)+{\partial }_{\overline{z}}(v\overline{w}),\\ & {\partial }_{\overline{z}}w=-3{\partial }_{z}v,\end{array}$

where the following standard notation of complex analysis is used: ${\partial }_z=\frac{1}{2}({\partial }_{x_1}-i{\partial }_{x_2})$, ${\partial }_{\overline{z}}=\frac{1}{2}({\partial }_{x_1}+i{\partial }_{x_2})$. The function $w$ here is an auxiliary function defined via $v$ up to a holomorphic summand. The function $v$ is generally assumed to be a real-valued function.

See also

• Integrable systems
• Nonlinear Schrödinger equation
• Nonlinear systems
• Davey–Stewartson equation
• Dispersive partial differential equation
• Kadomtsev–Petviashvili equation
• Korteweg–de Vries equation

References

• Kodam Y., Gibbons J. "Integrability of the dispersionless KP hierarchy"
• Zakharov V.E. "Dispersionless limit of integrable systems in 2+1 dimensions"
• Takasaki K., Takebe T. Rev. Math. Phys., 7, 743 (1995)
• Konopelchenko B.G. "Quasiclassical generalized Weierstrass representation and dispersionless DS equation", ArXiv: 0709.4148
• Konopelchenko B.G., Moro A. "Integrable Equations in Nonlinear Geometrical Optics", Studies in Applied Mathematics, 113(4), pp. 325–352 (2004)
• Dunajski M. "Interpolating integrable system". ArXiv: 0804.1234

External links

• Ishimori_system at the dispersive equations wiki