The Nonlinear Klein-Gordon Equation [chapter]

Vieri Benci, Donato Fortunato
2014 Springer Monographs in Mathematics  
7. ∂ 2 w ∂t 2 = ∂ 2 w ∂x 2 + f (w). Nonlinear Klein-Gordon equation. 1 • . Suppose w = w(x, t) is a solution of the nonlinear Klein-Gordon equation. Then the functions where C 1 , C 2 , and β are arbitrary constants, are also solutions of the equation (the plus or minus signs in w 1 are chosen arbitrarily). 2 • . Traveling-wave solution in implicit form: where C 1 , C 2 , k, and λ are arbitrary constants. 3 • . Functional separable solution: where C 1 and C 2 are arbitrary constants, and the
more » ... ction w = w(ξ) is determined by the ordinary differential equation See also special cases of the nonlinear Klein-Gordon equation: • Klein-Gordon equation with a power-law nonlinearity -1 , • Klein-Gordon equation with a power-law nonlinearity -2 , • modified Liouville equation , • Klein-Gordon equation with a exponential nonlinearity , • sinh-Gordon equation , • sine-Gordon equation .
doi:10.1007/978-3-319-06914-2_4 fatcat:dxbsn2abivclpg7knovqtdnvqq