Self-similar blowup solutions to the 2-component Degasperis–Procesi shallow water system

In this article, we study the self-similar solutions of the 2-component Degasperis–Procesi water system: (1) ρ t + k 2 u ρ x + ( k 1 + k 2 ) ρ u x = 0 , u t - u xxt + 4 uu x - 3 u x u xx - uu xxx + k 3 ρ ρ x = 0 . By the separation method, we can obtain a class of self-similar solutions, (2) ρ ( t , x ) = max f ( η ) a ( 4 t ) ( k 1 + k 2 ) / 4 , 0 , u ( t , x ) = a ˙ ( 4 t ) a ( 4 t ) x , a ¨ ( s ) - ξ 4 a ( s ) κ = 0 , a ( 0 ) = a 0 ≠ 0 , a ˙ ( 0 ) = a 1 , f ( η ) = k 3 ξ - η 2 k 3 ξ + α k 3 ξ 2 , where η = x a ( s ) 1 / 4 with s = 4 t ; κ = k 1 2 + k 2 - 1 , α ⩾ 0 , ξ < 0 , a0 and a1 are constants, which the local or global behavior can be determined by the corresponding Emden equation a(s), (2)2. The results are similar to the ones obtained for the 2-component Camassa–Holm equations. Our C0 solutions (2) can capture the evolution of breaking waves of that system. The constructed solutions could be applied to test the numerical computation for the system. In the last section, with the characteristic line method, blowup phenomenon for k3 ⩾ 0 is also studied.