On the asymptotic representation of surface waves in the form of two travelling Burgers waves (Q1363990)

The system \[ \psi_t+ (\psi\varphi)_x+ \mathbb{K}_1\psi+ \mathbb{K}_2\varphi=0,\;\varphi_t+{1\over 2} (\varphi^2)_x+ \mathbb{K}_3\psi+ \mathbb{K}_4\varphi= 0 \] is studied with respect to the unique solvability of the Cauchy problem. Asymptotic formulas for \(t\to\infty\) are given. Here, \(\psi\) and \(\varphi\) are real functions, \(x\in R_1\), \(t>0\). The symbols \(K_j(p)\) of the operators \(\mathbb{K}_j\), \[ \mathbb{K}_j\chi={1\over (2\pi)^2} \int_{R_2} e^{i(p,x)}K_j(p)\widehat\chi(p, t)dp \] are scalar complex functions.