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Painleve' analysis of a variable coefficient Sine-Gordon equation
Angelo Di Garbo1, Leone Fronzoni
1Dipartimento di Fisica dell'Universita di Pisa, Piazza Torricelli 2, 56100 Pisa, ItalyConsorzio Nazionale Interuniversitario di Fisica della Materia, Piazza Torricelli 2, 56100 Pisa, ItalyGruppo Nazionale di Struttura della Materia del Consiglio Nazionale delle Ricerche, Piazza Torricelli 2, 56100 Pisa, Italy.
Chaos (Woodbury, N.Y.)
|December 1, 1995
Summary
Researchers investigated the variable coefficient Sine-Gordon equation. They found it possesses the Painleve property if a specific nonlinear partial differential equation is satisfied, leading to new solutions.
Area of Science:
- Mathematical Physics
- Nonlinear Partial Differential Equations
Background:
- The Sine-Gordon equation is a fundamental model in nonlinear dynamics.
- Investigating the integrability of its variable coefficient variants is crucial for understanding complex physical phenomena.
Purpose of the Study:
- To determine the conditions under which the variable coefficient Sine-Gordon (vSG) equation exhibits the Painleve property.
- To derive specific forms of the coefficient function F(x,t) that ensure integrability.
Main Methods:
- Application of the Weiss, Tabor, and Carnevale (WTC) test for integrability.
- Analysis of the resulting nonlinear partial differential equation for the coefficient function F(x,t).
Main Results:
- The vSG equation possesses the Painleve property if F(x,t) satisfies a specific nonlinear PDE.
- The general solution for F(x,t) is found to be F(x,t) = F(1)(x+t)F(2)(x-t), where F(1) and F(2) are arbitrary functions.
- Particular solutions for the vSG equation were derived based on these findings.
Conclusions:
- The integrability of the vSG equation is directly linked to the structure of its coefficient function.
- The identified form of F(x,t) provides a pathway to generating a class of integrable vSG equations and their solutions.