In this paper we employ two recent analytical approaches to investigate the possible classesof traveling wave solutions of some members of a recently-derived integrable family of generalizedCamassa-Holm (GCH) equations.A recent, novel application of phase-plane analysis is employed to analyze the singular traveling waveequations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon, cuspon and compactonsolutions. Two of the GCH equations do not support singular traveling waves. The third equationsupports four-segmented, non-smooth $M$-wave solutions, while the fourth supports both solitary(peakon) and periodic (cuspon) cusp waves in different parameter regimes.Moreover, smooth traveling waves of the four GCH equations are considered. Here, we use a recenttechnique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbitsof their traveling-wave equations, corresponding to pulse and front (kink or shock) solutionsrespectively of the original PDEs. We perform many numerical tests in different parameter regime topinpoint real saddle equilibrium points of the corresponding GCH equations, as well as ensuresimultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic andheteroclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergentseries, high accuracy is attained with relatively few terms. We also show the traveling wave nature ofthesepulse and front solutions to the GCH NLPDEs.
|Numero di pagine||24|
|Rivista||COMMUNICATIONS IN NONLINEAR SCIENCE & NUMERICAL SIMULATION|
|Stato di pubblicazione||Published - 2014|
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics
Gambino, G., Choudhury, S. R., & Rehman (2014). Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations. COMMUNICATIONS IN NONLINEAR SCIENCE & NUMERICAL SIMULATION, 19, 1746-1769.