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Resonant interactions of nonlinear transverse waves in incompressible magnetoelasticity. (English) Zbl 0983.74038

Summary: An asymptotic expansion of high-frequency small-amplitude waves is applied to the equations of nonlinear elastic incompressible solid interacting with magnetic field. The one-dimensional case is discussed. First, we derive evolution transport equations of weakly nonlinear geomeric optics for the interacting nonlinear magnetoelastic waves. In contrast to the zero background field, the nonzero prestrain initial data (around which the asymptotics are applied) imply that some of the resonant interacting coefficients are nonzero. All such coefficients are calculated explicitly and analyzed for a certain nonzero constant state. Finally, we derive the equations for resonant triads.

MSC:

74J30 Nonlinear waves in solid mechanics
74F15 Electromagnetic effects in solid mechanics
74B20 Nonlinear elasticity
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