On the integrability of non-polynomial scalar evolution equations
The (symmetry) integrability of a nonlinear evolution partial differential equation (in 1+1 dimensions: time plus space) is considered tantamount to possession of an infinite number of symmetries. The equations under consideration are classified by a grading prescription (\(\lambda\)-homogeneity). In a previous paper the authors had ``produced the complete list of \(\lambda\)-homogeneous (symmetry) integrable equations with positive \(\lambda\)''. The main result of this paper is to ``give the complete list of 0-homogeneous integrable systems''. It turns out that such integrable equations are equivalent up to homogeneous transformations to the equations contained in hierarchies of: Heat Equation/potential Burgers equation \(u_t=u_2\); Potential modified Korteweg-de Vries equation, \(u_t=u_3+u^3_1\); Potential Kupershmidt equation, \(u_t=u_5+5u_2u_3-5u_1u^2_2+u^5_1\). (The reader will note that these equations are all polynomial; but nonpolynomial equations are also considered in the paper, justifying its title). The authors also ``draw some conclusions from these results for the case \(\lambda=-1\) which, although theoretically incomplete, seem to cover the known integrable systems for this case''.
scientific article; zbMATH DE number 1517169