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Self-similar blowup solutions to the 2-component Degasperis-Procesi shallow water system. (English) Zbl 1419.76097

Summary: We study the self-similar solutions of the 2-component Degasperis-Procesi water system: \[ \begin{cases} \rho_t+k_2u\rho_x+(k_1+k_2)\rho u_x=0,\\u_t-u_{xxt}+4uu_x-3u_xu_{xx}-uu_{xxx}+k_3\rho\rho_x=0.\end{cases}\tag{1} \](1) By the separation method, we can obtain a class of self-similar solutions \[ \begin{cases} \rho(t,x)=\max\left(\frac{f(\eta)}{a(4t)^{k_1-k_2)/4}},0\right),\quad u(t,x)=\frac{\dot a(4t)}{a(4t)}x,\\ \ddot a(s)-\frac{\xi}{4a(s)^K}=0, \quad a(0)=a_0\neq 0,\quad \dot a(0)=a_1,\\ f(\eta)-k_3\xi\sqrt{-\frac{\eta^2}{k_3\xi}+\left(\frac{\alpha}{k_3\xi}\right)^2}.\end{cases}\tag{2} \] where \(\eta=\frac{x}{a(s)^{1/4}}\) with \(s=4t\); \(\kappa=\frac{k_1}{2} +k_2-1\), \(\alpha\geqslant 0\), \(\xi<0\), \(a_0\) and \(a_1\) 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 \(C^{0}\) 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 \(k_{3} \geqslant 0\) is also studied.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
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References:

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