Exact Peakon, Compacton, Solitary Wave, and Periodic Wave Solutions for a Generalized KdV Equation

Figure 2

Bifurcation sets and phase portraits of system (7) when 𝛼<0.

(a) 𝑐<−𝛽/2𝛼,𝑔<−𝑐²/2

(b) 𝑐<−𝛽/2𝛼,𝑔=−𝑐²/2

(c) 𝑐<−𝛽/2𝛼,−𝑐²/2<𝑔<−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²

(d) 𝑐<−𝛽/2𝛼,𝑔=−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²

(e) 𝑐<−𝛽/2𝛼,−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼²

(f) 𝑐<−𝛽/2𝛼,𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼²

(g) 𝑐<−𝛽/2𝛼,𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼²

(h) 𝑐>−𝛽/2𝛼,𝑔<−𝑐²/2

(i) 𝑐>−𝛽/2𝛼,𝑔=−𝑐²/2

(j) 𝑐>−𝛽/2𝛼,−𝑐²/2<𝑔<−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²

(k) 𝑐>−𝛽/2𝛼,𝑔=−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²

(l) 𝑐>−𝛽/2𝛼,−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼²

(m) 𝑐>−𝛽/2𝛼,𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼²

(n) 𝑐>−𝛽/2𝛼,𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼²