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Conservation laws for a nonlinear wave equation. (English) Zbl 0892.35098

Summary: This paper is concerned with a nonlinear wave equation which has been proposed as a model for conduit flow and migration of magma, namely \[ \varphi_t + (\varphi^n +m\varphi^{n-m-1} \varphi_t\varphi_z -\varphi^{n-m} \varphi_ {tz})_z =0. \] A systematic search for conservation laws is performed, allowing all possible values of the two parameters \(n\) and \(m\). In most cases, it is demonstrated that there exist precisely two independent laws and these are exhibited. However, two special cases \(m=1\), \(n\neq 0\) and \(m= n+1\), \(n\neq 0\) are discovered for which the proof is inconclusive and the second of these leads to a third law for a single parameter family.

MSC:

35L65 Hyperbolic conservation laws
35Q53 KdV equations (Korteweg-de Vries equations)
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