Anomalous exponents and dipole solutions for the thin film equation (Q2783705)

In this paper the authors use methods from the theory of ODEs and formal asymptotics to construct special solutions for the moving boundary problem generated by the thin-film equations, NEWLINE\[NEWLINE h_t + (|h|^n h_{xxx})_x =0, NEWLINE\]NEWLINE on the half line \(x>0\), with the boundary conditions NEWLINE\[NEWLINE h=h_x =0 \text{ at } x=0, x=s(t) \text{ and } \lim_{x\rightarrow s(t)_-} h^n h_{xxx}=0, NEWLINE\]NEWLINE where \(s(t)\) is the location of the moving boundary. The main part of the paper concerns the regime \(0<n<2\). Motivation for considering such boundary conditions is given in Appendix 1. Results obtained by these methods are then strengthened by numerics. NEWLINENEWLINENEWLINEMore precisely, solutions of the form \(h(x,t)=t^{-\alpha} f(x/t^\beta)\), \(n\alpha+4\beta=1\), are sought, so that \(f(\eta) \geq 0\) satisfies the fourth-order ODE NEWLINE\[NEWLINE (f^n f''')'= \beta \eta f' +\alpha f , \quad 0 < \eta < \eta_0, \tag{1}NEWLINE\]NEWLINE with the boundary conditions NEWLINE\[NEWLINE f(0)=f'(0)=f(\eta_0)=f'(\eta_0)=f(\eta_0)^n f'''(\eta_0)=0. NEWLINE\]NEWLINE The solutions sought are connections between two critical points of various dynamical systems derivable from (1). Such solutions are termed dipoles. NEWLINENEWLINENEWLINEAnalysis of the behaviour of the dynamical systems obtained by various changes of variables in (1) (including a modified Poincaré transformation) in a neighbourhood of critical points provides information on possible asymptotic behaviour of the dipole solutions as \( \eta \rightarrow \infty\). This ODE analysis together with numerics leads to a graph of exponents \(\alpha\) versus \(n\); see Figure 3.9. NEWLINENEWLINENEWLINEExact dipole solutions are known in the case \(n=1\), \(n=2\). Hence Section 4 presents asymptotic expansions in a deviation of the exponent \(n\) from these two values. The limit \(n \rightarrow -\infty\) is also considered. NEWLINENEWLINENEWLINEThe resulting picture is that of a rich variety of self-similar solutions of the second kind which invites further investigation.