Small Fluctuations in $\lambda {\varphi }^{n+1}$ Theory in a Finite Domain: An Hirota's Method Approach

We present a method to calculate small stationary fluctuations around static solutions describing bound states in a $(1+1)$-dimensional $\lambda {\varphi }^{n+1}$ theory in a finite domain. We also calculate explicitly fluctuations for the $\lambda {\varphi }^4$. These solutions are written in terms of Jacobi Elliptic functions and are obtained from both linear and nonlinear equations. For the linear case we get eingenvalues of a Lam'e type Equation and the nonlinear one relies on Hirota’s Method.