Generalized Korteweg-de Vries equation

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Half-line theory

The gKdV Cauchy-boundary problem on the half-line is

u_t + u_{xxx} + u_x + u^k u_x = 0; u(x,0) = u_0(x); u(0,t) = h(t)

The sign of u_{xxx} is important (it makes the influence of the boundary x=0 mostly negligible), the sign of u u_x is not. The drift term u_x is convenient for technical reasons; it is not known whether it is truly necessary.

  • LWP is known for initial data in H^s and boundary data in H^{(s+1)/3} when s > 3/4 [CoKn-p].

Miscellaneous gKdV results

[Thanks to Nikolaos Tzirakis for some corrections - Ed.]