Construction of soliton equations using special polynomials

A simple, algorithmic approach is proposed for the construction of the most general family of equations of a given scaling weight, possessing, at least, the same single-soliton solution as a given, lower scaling weight equation. The construction exploits special polynomials–differential polynomials in the solution, u, of an evolution equation, which vanish identically when u is a single-soliton solution. Applying the approach to different types of evolution equations yields new results concerning the most general families of evolution equations in a given scaling weight, which possess solitary wave solutions. The same method can be applied in the identification of families of evolution equations of mixed scaling weight (and, in general, of any structure), which admit single-soliton solutions of a desired form.