Studies in Applied Mathematics

Ablowitz, M. J.; Ladik, J. F.

doi: 10.1002/sapm19775711pmid: N/A

A systematic method for isolating certain nonlinear partial difference equations which are solvable by the method of inverse scattering is discussed. The equations are characterized by the dispersion relation (amplification factor) of the associated linearized equation. Corresponding to every neutrally stable linear scheme is a nonlinear difference equation which converges to an exactly solvable nonlinear evolution equation in the continuum limit. Explicit solition solutions are given.

Ablowitz, M. J.; Segur, H.

doi: 10.1002/sapm197757113pmid: N/A

The long‐time asymptotic solution of the Korteweg‐deVries equation, corresponding to initial data which decay rapidly as |x|→∞ and produce no solitons, is found to be considerably more complicated than previously reported. In general, the asymptotic solution consists of exponential decay, similarity, rapid oscillations and a “collisionless shock” layer. The wave amplitude in this layer decays as ((lnt)/t)2/3. Only for very special initial conditions is the shock layer absent from the solution.

Greenspan, H. P.

doi: 10.1002/sapm197757145pmid: N/A

The surface tension of a small viscous droplet is made variable by the deposition of a surface active material. The resultant shear forces produce a deformation of and a fluid motion within the droplet. The process, including the nonlinear convection or spread of surfactant, is examined theoretically. Some experimental results are reported, and application to cell division is discussed.

Davis, Stephen H.; Rosenblat, S.

doi: 10.1002/sapm197757159pmid: N/A

A study is made of differential equations which have low‐frequency periodic bifurcating solutions. It is shown that the classical Poincaré‐Hopf technique for constructing periodic solutions encounters difficulties in the low‐frequency limit, and that the branches are determined by nonlinear, rather than linear, balances. Two types of models are investigated: one is autonomous and the other non‐autonomous, with a forcing term of small, fixed frequency. The latter model is relevant to several fluid‐dynamical situations.

Rubenfeld, L. A.

doi: 10.1002/sapm197757177pmid: N/A

An asymptotic procedure for deriving equations governing the passage of a weakly coupled nonlinear system of oscillators is discussed. The procedure avoids an inner‐outer‐matching technique and is valid when the small coupling and detuning parameters are arbitrary. Resonance is permitted to occur at one or several instances of time or to last for a finite length of time. Numerical results are discussed.