Results 1 - 10 of 4434
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[en] In this paper, first we introduce four novel extensions of Zakharov–Kuznetsov equations which are their logarithmic forms. Then, we investigate the new logarithmic equations for their Gaussian solitary waves. After that, we obtain Gaussian solitons for all models. We show that all logarithmic models are characterized by their Gaussian solitary waves. These extensions can conduct interested researchers to obtain logarithmic extensions for another equations. Besides, the presented Gaussian solitary waves can be useful both mathematically and physically. (paper)
[en] In this paper, we use the generalized semiflow theory to study the longtime dynamical properties for a class of semilinear hyperbolic equations. The existence of global attractors is shown for the equations with no Lipschitz continuity assumption on their nonlinear terms. The results obtained here are generalizations of the related ones in Ball [Ball JM. Global attractors for damped semilinear wave equations. Discret Contin Dyn Syst 2004;10:31-52]
[en] The harmonic wave equation in inhomogeneous media is exactly split into coupled first-order equations with respect to a principal direction of propagation according to the Bremmer scheme. The resulting one-way wave equation is shown not to conserve energy flux for dimensions two and three against the general belief in one-way wave propagation or parabolic equation literature. Conservation of energy flux is only ensured in the high frequency limit. On the other hand, a simple invariant is found that may be seen as a generalization of the Snell law to arbitrary, non-stratified, media. Similarly, the reciprocity property is not fully ensured in general and the time-reversal symmetry is ensured for propagating fields. Besides, in the one-way wave equation, the additional term to the standard parabolic equation is shown to strengthen mode coupling. The analysis encompasses the evanescent waves
[en] A sequence of numerically tractable higher-order parabolic approximations is derived for the reduced wave equation in an inhomogeneous medium. The derivation is motivated by a definition of waves propagating in a distinguished direction. For a homogeneous medium these definitions are exact and yield uncoupled, infinite-order parabolic equations which are equivalent to the wave equation. The difficulty of obtaining higher-order parabolic approximations for the elastic wave equation in an inhomogeneous medium is also discussed
[en] The convergence of certain semidiscrete approximation schemes based on the velocity-stress formulation of the wave equation and spaces such as those introduced by Raviart and Thomas is discussed. The discussion also applies to similar schemes for the equations of elasticity
[en] In this work, we determine the critical exponent for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower-order terms, when these terms make both equations in some sense “parabolic-like.” For the blow-up result, the test functions method is applied, while for the global existence (in time) results, we use – estimates with additional regularity.
[en] We consider a system of two coupled nonconservative wave equations. For this system, we prove several observability estimates. Those observability estimates are sharp in the sense that they lead by duality to the controllability (exact or approximate) of the coupled system with a single control acting through one of the equations only while keeping the same controllability time as for a single equation. Existing results in the literature either require two controls, or in the case of a single control, they have a controllability time that blows up as the coupling parameter goes to zero. Our proofs rely on: (i) Carleman estimates, (ii) energy estimates and (iii) localizing arguments. The results obtained complement and improve, in some sense, earlier results while at the same time providing new uniqueness results.