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

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].

### 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