Fig. 2a. Solution to WP1130 Cauchy problem for Eq. (10) with the initial conditions of type (21) for three successive moments t=0,0.5,1.0t=0,0.5,1.0 at the following values of input parameters β=1β=1, ηl=1.25,ηr=1.0ηl=1.25,ηr=1.0. Initial film thicknesses h0h0 and h6h6 in front and behind the shock wave front, respectively.Figure optionsDownload full-size imageDownload as PowerPoint slide

Fig. 2b. Solution to the Cauchy problem for Eq. (10) with the initial conditions of type (21) for three successive moments t=0,0.6,1.5t=0,0.6,1.5 at the following values of input parameters β=-1β=-1, ηl=2.0,ηr=1.3ηl=2.0,ηr=1.3. Initial film thicknesses h0h0 and h6h6 in front and behind the shock wave front, respectively.Figure optionsDownload full-size imageDownload as PowerPoint slide

Evolution of generalized solution (21) at β>0β>0 is shown in Fig. 2(a), and at β<0β<0 companion cells is shown in Fig. 2(b). At β>0β>0, solution (21) was obtained for all t>0t>0. The propagation velocity of its shock D(H(t,ηl),H(t,ηr))D(H(t,ηl),H(t,ηr)) increases with time, and the amplitude of this shock decrease, and this follows from formulaequation(23)H(t,ηl,β)-H(t,ηr,β)=ηl2-ηr2H(t,ηl,β)+H(t,ηr,β).