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

TL;DR

This paper develops a method to analyze small stationary fluctuations around static solutions in a finite domain for the $ ext{lambda} \phi^{n+1}$ theory, explicitly calculating fluctuations for the $ ext{lambda} \phi^4$ case using Jacobi Elliptic functions and Hirota's Method.

Contribution

It introduces a novel approach combining linear and nonlinear analysis, including Hirota's Method, to compute fluctuations in finite domain $ ext{lambda} \phi^{n+1}$ theories.

Findings

Eigenvalues of Lamé type equations obtained for linear fluctuations.

Explicit fluctuation solutions for $ ext{lambda} \phi^4$ theory expressed with Jacobi Elliptic functions.

Demonstrated the effectiveness of Hirota's Method in nonlinear fluctuation analysis.

Abstract

We present a method to calculate small stationary fluctuations around static solutions describing bound states in a $(1+1)$-dimensional $\lambda \phi^{n+1}$ theory in a finite domain. We also calculate explicitly fluctuations for the $\lambda \phi^4$. 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.