Asymptotic representation of surface waves in the form of two traveling Burgers waves (Q1916668)

The authors study the Cauchy problem for the system \[ \psi_t+(\psi\varphi)_x+ K_{11}\psi+K_{12}\varphi=0, \qquad \varphi_t+0,5(\varphi^2)_x+ K_{21}\psi+K_{22}\varphi=0, \] where \(\psi\) and \(\varphi\) are real functions, \(x\in \mathbb{R}^1\), \(t>0\), and \(K_{ij}\) are linear pseudodifferential operators defined via the inverse Fourier transform. This system covers numerous well-known systems from mechanics; in particular, a special choice of operators \(K_{ij}\) provides a system describing surface water waves with allowance for viscosity and surface tension. Under some assumptions on the eigenvalues of the matrix \((K_{ij})\) it is proved that for \(t\to \infty\) uniformly with respect to \(x\) \[ \psi(x,t)= b\nu_1(x,t)+ b\nu_2(x,t)+ O(t^{-0,5-\gamma}), \qquad \varphi(x,t)= -\nu_1(x,t)+ \nu_2(x,t)+ O(t^{-0,5-\gamma}), \] where \(b\) and \(\gamma>0\) are constants and \(\nu_i(x,t)\) are solutions of the Cauchy problem for the system of Burgers' equations with transfer.