Resonance-based schemes for dispersive equations via decorated trees

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.

ON THE INSTABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS—REVISITED

The purpose of this paper is to present an extension of the results in [8]. We establish a more general proof for the moving kernel formula to prove the spectral stability of periodic traveling wave solutions for the regularized Benjamin–Bona–Mahony type equations. As applications of our analysis, we show the spectral instability for the quintic Benjamin–Bona–Mahony equation and the spectral (orbital) stability for the regularized Benjamin–Ono equation.

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

The solutions of the Cauchy problem of the KdV equation on a periodic domain $\T$, \[ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in \T, \ t\in \R,\] possess neither the sharp Kato smoothing property, \[ \phi \in H^s (\T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (\T, L^2 (0,T)),\] nor the Kato smoothing property, \[ \phi \in H^s (\T) \implies u\in L^2 (0,T; H^{s+1} (\T)).\] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain $\T$, \[ u_t +uu_x +u_{xxx} - g(x) (g(x) u)_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in \T, \ t>0 \, ,\ \qquad (1) \] where $g\in C^{\infty} (\T)$ is a real value function with the support \[ \mbox{$\omega = \{ x\in \T, \ g(x) \ne 0\}$.}\] It is shown that \begin{itemize} \item[(1)] if $\omega\ne \emptyset$, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; \item[(2)] if $g$ is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property. \end{itemize}

Initial-Boundary Value Problems for Nonlinear Dispersive Equations of Higher Orders Posed on Bounded Intervals with General Boundary Conditions

The present article concerns general mixed problems for nonlinear dispersive equations of any odd-orders posed on bounded intervals. The results on existence, uniqueness and exponential decay of solutions are presented.