Abstract
We study the generalized bosonic string equation
on Euclidean space \({\mathbb {R}^n}\) . First, we interpret the nonlocal operator \({\Delta e^{-c\,\Delta}}\) using entire vectors of Δ in \({L^2(\mathbb{R}^n)}\) , and we show that if \({U(x, \phi) = \phi(x) + f(x)}\) , in which \({f \in L^2(\mathbb {R}^n)}\) , then there exists a unique real-analytic solution to the Euclidean bosonic string in a Hilbert space \({{\mathcal H}^{c,\infty}(\mathbb {R}^n)}\) we define precisely below. Second, we consider the case in which the potential \({U(x, \phi)}\) in the generalized bosonic string equation depends nonlinearly on \({\phi}\) , and we show that this equation admits real-analytic solutions in \({{\mathcal H}^{c,\infty}(\mathbb R^n)}\) under some symmetry and growth assumptions on U. Finally, we show that the above given equation admits real-analytic solutions in \({{\mathcal H}^{c,\infty}(\mathbb {R}^n)}\) if \({U(x, \phi)}\) is suitably near \({U_0(x, \phi) = \phi}\) , even if no symmetry assumptions are imposed.
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Communicated by Palle Jorgensen.
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Górka, P., Prado, H. & Reyes, E.G. Nonlinear Equations with Infinitely many Derivatives. Complex Anal. Oper. Theory 5, 313–323 (2011). https://doi.org/10.1007/s11785-009-0043-z
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DOI: https://doi.org/10.1007/s11785-009-0043-z