Abstract
We further generalize the generalized short pulse equation studied recently in [Commun. Nonlinear Sci. Numer. Simulat. 39 (2016) 21-28; arXiv:1510.08822], and find in this way two new integrable nonlinear wave equations which are transformable to linear Klein-Gordon equations.
Highlights
We study the integrability of the nonlinear wave equation uxt = au2uxx + buu2x containing two arbitrary parameters, a and b, not equal zero simultaneously
We show that this equation (1) is integrable in two distinct cases, namely, when a/b = 1/2 and a/b = 1, which correspond via scale transformations of variables to the equations uxt = 1 u3 (2)
In [1], we studied the integrability of the generalized short pulse equation uxt = u + au2uxx + buu2x containing two arbitrary parameters, a and b, not equal zero simultaneously
Summary
We study the integrability of the nonlinear wave equation uxt = au2uxx + buu2x (1). There is only one essential parameter in (1), the ratio a/b or b/a, which is invariant under the scale transformations of u, x and t, while the values of a and b are not invariant We show that this equation (1) is integrable in two (and, most probably, only two) distinct cases, namely, when a/b = 1/2 and a/b = 1, which correspond via scale transformations of variables to the equations uxt = 1 u3.
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