277 research outputs found
Convergent Numerical Schemes for the Compressible Hyperelastic Rod Wave Equation
We propose a fully discretised numerical scheme for the hyperelastic rod wave
equation on the line. The convergence of the method is established. Moreover,
the scheme can handle the blow-up of the derivative which naturally occurs for
this equation. By using a time splitting integrator which preserves the
invariants of the problem, we can also show that the scheme preserves the
positivity of the energy density
Periodic conservative solutions for the two-component Camassa-Holm system
We construct a global continuous semigroup of weak periodic conservative
solutions to the two-component Camassa-Holm system, and , for
initial data in . It is
necessary to augment the system with an associated energy to identify the
conservative solution. We study the stability of these periodic solutions by
constructing a Lipschitz metric. Moreover, it is proved that if the density
is bounded away from zero, the solution is smooth. Furthermore, it is
shown that given a sequence of initial values for the densities that
tend to zero, then the associated solutions will approach the global
conservative weak solution of the Camassa-Holm equation. Finally it is
established how the characteristics govern the smoothness of the solution.Comment: To appear in Spectral Analysis, Differential Equations and
Mathematical Physics, Proc. Symp. Pure Math., Amer. Math. So
Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics
We show existence of a global weak dissipative solution of the Cauchy problem
for the two-component Camassa-Holm (2CH) system on the line with nonvanishing
and distinct spatial asymptotics. The influence from the second component in
the 2CH system on the regularity of the solution, and, in particular, the
consequences for wave breaking, is discussed. Furthermore, the interplay
between dissipative and conservative solutions is treated.Comment: arXiv admin note: text overlap with arXiv:1111.318
Convergent numerical schemes for the compressible hyperelastic rod wave equation
We propose a fully discretised numerical scheme for the hyperelastic rod wave equation on the line. The convergence of the method is established. Moreover, the scheme can handle the blow-up of the derivative which naturally occurs for this equation. By using a time splitting integrator which preserves the invariants of the problem, we can also show that the scheme preserves the positivity of the energy densit
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