Non-analytic solutions of nonlinear wave models. (English) Zbl 0940.35176

Li, Yi A.; Olver, Peter J.; Rosenau, Philip

Grosser, Michael (ed.) et al., Nonlinear theory of generalized functions. Proceedings of the workshop on nonlinear theory of nonlinear functions, Erwin-Schrödinger-Institute, Vienna, Austria, October-December 1997. Boca Raton, FL: Chapman & Hall. Chapman Hall/CRC Res. Notes Math. 401, 129-145 (1999).

An historical survey of the recent results concerning non-analytic solutions of nonlinear wave models is given, especially concerning Boussinesq’s models in the equation \(u_t+u_x+(u^2)_x+u_{xxx}=0\), representing the Korteweg-de Vries equation, which models the weakly nonlinear regime in which nonlinear effects are balanced by dispersion.
The trajectories and graphs of peakon, loopon and compacton solutions of the \(K(2,2)\) equation \(u_t \pm (u^2)_x +(u^2)_{xxx}=0\), respectively trajectories and graphs of the analytic and tipon solutions of the \(K(3,3)\) equation are given.
The authors present also the (associated to KdV equation) Camassa-Holm equation \(u_t+u_x-\nu u_{xxt}+3(u^2)_x-\nu (u^2)_{xxx}+\nu (u_x^2)_x=0\) which was applied to systematically additional nonlinearly dispersive, integrable systems associated with many other classical weakly nonlinear soliton equations.
As a particularly interesting example the generalized Boussinesq system is presented: \[ v_t+2w_t-2v_{xt} = w_x-v_{xx}+(\frac{1}{2}v^2+vw)_x-(\frac{1}{2}v^2)_{xx}, \]
\[ w_t+2w_{xt}=w_{xx}+(vw+\frac{1}{2}w^2+vw_x)_x. \] In all the cases one looks for solutions of the form \(\varphi(x-ct)\) or \(\varphi(x+ct)\) with \(c\) representing the speed of traveling wave.
For the entire collection see [Zbl 0918.00026].

Reviewer: P.Craciunas (Iaşi)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems

Keywords:

Boussinesq models; KdV equation; Camassa-Holm equation; traveling wave