On The Stability of Solutions of a Nonlinear Field Equation.
Davis, William Joseph
Woehler, Karlheinz E.
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Solutions to a nonlinear wave equation were analyzed for their stability. The wave equation is a Klein-Gordon equation with the mass replaced by the square of the wave function. This wave equation has propagating solutions which are unbounded or periodic, depending on the sign of the nonlinear term and the propagation speed which can be sub- or super-light velocity. The stability of the Deriodic sublight velocity solution was investigated by the method of characteristic exponents and was found to be indifferent. Liapounoff's direct method and Sturrock's analysis of the dispersion relation combined with a WKB technique were applied to a linearized perturbation on a static solution of 2 the field equation. The periodic solution with (3 < 1 is stable, while the method of characteristic exponents gives indifference. The super-light velocity solutions are unstable. Due to the limitations of the approximations, it could not be determined whether the instabilitv is absolute or convective.
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