Pulse Equation Research Articles

Continuous and jumping stationary transitions of a two-speed multiphase medium from one state to another

A model of two-velocity multiphase medium with equal phase pressures was investigated. Two pulse equations were used where interphase force has a stabilizing term depending on flow parameters gradient. Phases were assumed incompressible. It was shown that this asssumption does not have any impact on the main conclusions while the model is significantly simplified. One does not need any energy equations because the movement does not depend on phase temperatures. Two differential equations describing relative phase movement were derived. Each of the two differential equations describing relative phase movement leads to its own wave velocity. For a steady-state stable transition when the mixture parameters do not change with time in some coordinate system (a steady-state or automodel flow) wave velocity must be equal for all the differential equations. This gives a condition for calculation of the stabilizing term. We defined conditions for continuous transition from one state to another as well as conditions when this transition can be only discontinuous. Analytical solutions helped us to find stabilizing term for interphase friction force (the only posssible in the corresponding class).

Periodic Traveling Waves of the Regularized Short Pulse and Ostrovsky Equations: Existence and Stability

We construct various periodic traveling wave solutions of the Ostrovsky/Hunter--Saxton/short pulse equation and its KdV regularized version. For the regularized short pulse model with small Coriolis parameter, we describe a family of periodic traveling waves which are a perturbation of appropriate KdV solitary waves. We show that these waves are spectrally stable. For the short pulse model, we construct a family of traveling peakons with corner crests. We show that the peakons are spectrally stable as well.

Reply to “Comment on `On Two-Loop Soliton Solution of the Schäfer–Wayne Short- Pulse Equation Using Hirota's Method and Hodnett–Moloney Approach' ”

Following a recent comment paid by Zhang and Li to the paper entitled ‘‘On Two-Loop Soliton Solution of the Schafer–Wayne Short-Pulse Equation Using Hirota’s Method and Hodnett–Moloney Approach’’, we show that solving initial value problems may lead to solutions different from that of vanishing boundary value problems. In a recently published work, we have investigated the two-loop soliton solutions to the Schafer–Wayne short pulse (SWSP) equation given by