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Zhuravlev, V. M.

Multidimensional nonlinear wave equations with multivalued solutions. (English. Russian original) Zbl 1282.35248

Theor. Math. Phys. 174, No. 2, 236-246 (2013); translation from Teor. Mat. Fiz. 174, No. 2, 272-284 (2013).
Summary: We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d’Alembert equations, nonlinear Klein-Gordon equations, and nonlinear telegraph equations.

Cited in 5 Documents

MSC:

35L72 Second-order quasilinear hyperbolic equations
35Q60 PDEs in connection with optics and electromagnetic theory

Keywords:

exact solution of multidimensional hyperbolic equations; breaking wave; medium with nonlinear polarization; d’Alembert equations; nonlinear Klein-Gordon equations; nonlinear telegraph equations
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References:

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