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New exact solutions for the classical Drinfel’d-Sokolov-Wilson equation. (English) Zbl 1181.35221

Summary: We investigate the classical Drinfel’d-Sokolov-Wilson equation (DSWE)

${u}_{t}+pv{v}_{t}=0,\phantom{\rule{2.em}{0ex}}{v}_{t}+ru{v}_{x}+s{u}_{x}v+q{v}_{xxx}=0,$

where $p,q,r,s$ are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended.

##### MSC:
 35Q51 Soliton-like equations 35C08 Soliton solutions of PDE 35B10 Periodic solutions of PDE 35B44 Blow-up (PDE) 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 37K50 Bifurcation problems (infinite-dimensional systems)
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