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Eigenvalues, and instabilities of solitary waves. (English) Zbl 0776.35065
The authors study an eigenvalue problem associated to systems of linear ordinary differential equations with asymptotically constant coefficients by using the analytic function $$D(\lambda)$$ introduced by Evans (1975). It is developed a general theory of $$D(\lambda)$$ that classifies the role of the essential spectrum in applications. They establish new integral formulae for the derivatives of $$D(\lambda)$$ that are used to study the linear exponential instability of the solitary waves of the generalized Korteweg-de Vries (KdV), Benjamin-Bona-Mahoney (BBM) and regularized Boussinesq equations.
More precisely, it is shown that for all the above three equations, when the instability condition $${d\over dc}{\mathcal N}[u_ c]<0$$ holds (being $$u_ c(x,t)=u_ c(x-ct)$$ the solitary wave and $${\mathcal N}(u)$$ a generalized momentum or impulse functional associated with the translation-invariant Hamiltonian structure of the equation), a real unstable eigenvalue exists with $$\lambda>0$$, for the linearized evolution equation around $$u_ c$$. This eigenvalue gives rise to a non-oscillatory and exponentially growing solution of the linearized evolution equation. For the gKdV and gBBM equation, it is also shown that there is at most one eigenvalue with $$\text{Re} \lambda>0$$.
Finally we must remark that the method used in this paper does not require the characterization of the solitary wave as a critical point of a modified Hamiltonian functional, whose second variation has a finite- dimensional negative subspace, as it happens in the methods used in previous works.