Nonlinear Hamiltonian equations with fractional damping (Q2737923)

The authors deal with the fractional differential equation NEWLINE\[NEWLINED^2u+CD^\lambda u+F(u)=0, \quad D=\frac{d}{dx}, \qquad 0\leq t\leq T, \tag{1}NEWLINE\]NEWLINE where \(C\) is a constant and \(D^{\lambda}\) denotes the Caputo or Riemann-Liouville fractional derivative of order \(\lambda\), \(0<\lambda \leq 2\). When \(C=0\), the equation (1) is Hamiltonian with the Hamiltonian function \(H(u,v)=V(u)+v^{2}/2\), where \(V(u)\) is a primitive function of \(F(u)\). The authors prove the existence and the uniqueness of a solution \(u(t,\lambda)\in C^{2}[0,T]\) to the initial boundary problem \(u(0+)=u_{0}\), \(Du(0+)=v_{0}\), for equation (1) with the Caputo fractional derivative \(D^{\lambda}\). It is shown that for \(0<\lambda \leq 1\) such a solution satisfies a certain dissipation condition. It is also established that this solution is an analytic function of \(\lambda\) when \(0<\lambda <1\) and \(1<\lambda <2\). It is proved that solutions to equation (1) with the Riemann-Liouville derivative \(D^{\lambda}\) do not belong to \(C^{2}[0,T]\) and their second-order derivatives are unbounded as \(t\to 0\). Numerical examples are given.