Traveling wave solutions of degenerate coupled multi-KdV equations

Traveling wave solutions of degenerate coupled ℓ-KdV equations are studied. Due to symmetry reduction these equations reduce to one ordinary differential equation (ODE), i.e., (f′)2 = Pn(f) where Pn(f) is a polynomial function of f of degree n = ℓ + 2, where ℓ ≥ 3 in this work. Here ℓ is the number of coupled fields. There is no known method to solve such ordinary differential equations when ℓ ≥ 3. For this purpose, we introduce two different types of methods to solve the reduced equation and apply these methods to degenerate three-coupled KdV equation. One of the methods uses the Chebyshev’s theorem. In this case, we find several solutions, some of which may correspond to solitary waves. The second method is a kind of factorizing the polynomial Pn(f) as a product of lower degree polynomials. Each part of this product is assumed to satisfy different ODEs.