On the structure of spectra of modulated travelling waves (Q2774474)

The system of reaction-diffusion equations NEWLINE\[NEWLINEu_t=Du_{xx}+f(u),\;x\in \mathbb R\tag{1}NEWLINE\]NEWLINE is considered, where \(D\) is a diagonal matrix, as well as NEWLINE\[NEWLINE u_t=u_{xx}+\Delta u+f(u), \quad(x,y)\in \mathbb R\times \Omega ,\tag{2}NEWLINE\]NEWLINE \(\Omega \subset \mathbb R^n\), \(\Delta \) Laplace operator in \(y\). In the first case, the structure of the spectra for modulated waves, i.e., solutions \(\widetilde q(x,t)\) of (1) satisfying \(\widetilde q(x,t+T)=\widetilde q(x-cT,t)\) for some \(c\) and \(T\), are investigated. The problem is linearized at this solution, and the spectrum of the linearization is characterized. The associated first-order system of this linearization is denoted by \(\Phi \) on \(L^2(\mathbb R,\mathbb C^n)\): NEWLINE\[NEWLINEv_t=Dv_{\xi\xi}+cv_\xi+a(\xi,t)v.NEWLINE\]NEWLINE The associated first-order system leads to the operator \(\mathcal T\). In Theorem 2.6 it is shown that \(\mathcal T\) is Fredholm if and only if NEWLINE\[NEWLINEv_\xi=w,\;w_\xi=D^{-1}(v_t+\alpha v-cw-a(\xi,t)v)NEWLINE\]NEWLINE has an exponential dichotomy. This leads to a characterization of the point spectrum, essential spectrum and pure point spectrum of \(\Phi \) via exponential dichotomies. These results are applied to long-wavelength periodic waves, in which case more explicit formulas of the spectrum are derived.