Search

In this paper, we consider several time-fractional generalizations of the Jordan–Moore–Gibson–Thompson (JMGT) equations in nonlinear acoustics as well as their linear Moore–Gibson–Thompson (MGT) versions. Following the procedure described in Jordan (2014), these time-fractional acoustic equations are derived from four fractional versions of the Maxwell–Cattaneo law in Compte and Metzler (1997). Additionally to providing well-posedness results for each of them, we also study the respective limits as the fractional order tends to one, leading to the classical third order in time (J)MGT equation. [ABSTRACT FROM AUTHOR]

In this paper, we use the double Wronskian reduction technique to consider nonlocal reductions of a nonisospectral (2+1)-dimensional breaking Ablowitz--Kaup--Newell--Segur equation. Various types of solutions, including soliton-like solutions and Jordan-block solutions, for the resulting nonlocal equations are derived. Dynamics of these obtained solutions are analyzed and illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we study large-time behaviors for a fundamental model in nonlinear acoustics, precisely, the viscous Jordan–Moore–Gibson–Thompson (JMGT) equation in the whole space R n. This model describes nonlinear acoustics in perfect gases under irrotational flow and equipping Cattaneo's law of heat conduction. By employing refined WKB analysis and Fourier analysis, we derive first- and second-order asymptotic profiles of solution to the Moore–Gibson–Thompson (MGT) equation as t ≫ 1 , which illustrates novel optimal estimates for the solutions even subtracting its profiles. Concerning the nonlinear JMGT equation, via suggesting a new decomposition of nonlinear portion, we investigate the existence and large-time profiles of global (in time) small data Sobolev solutions with suitable regularity. These results help bridge a new connection between the JMGT equation and diffusion-waves as t ≫ 1. [ABSTRACT FROM AUTHOR]

We develop a Cauchy matrix reduction technique that enables us to obtain solutions for the reduced local and nonlocal complex equations from the Cauchy matrix solutions of the original nonreduced systems. Specifically, by imposing local and nonlocal complex reductions on some Ablowitz–Kaup–Newell–Segur-type equations, we study some local and nonlocal complex equations involving the local and nonlocal complex modified Korteweg–de Vries equation, the local and nonlocal complex sine-Gordon equation, the local and nonlocal potential nonlinear Schrödinger equation, and the local and nonlocal potential complex modified Korteweg–de Vries equation. Cauchy matrix-type soliton solutions and Jordan block solutions for the aforesaid local and nonlocal complex equations are presented. The dynamical behavior of some of the obtained solutions is analyzed with graphical illustrations. [ABSTRACT FROM AUTHOR]

We study local and nonlocal complex reductions of a discrete negative-order Ablowitz–Kaup–Newell–Segur equation. For the resulting local and nonlocal complex discrete sine-Gordon equations, we construct solutions of the Cauchy matrix type, including soliton solutions and Jordan-block solutions. The dynamics of -soliton solutions are analyzed and illustrated. Continuum limits of the resulting local and nonlocal complex discrete sine-Gordon equations are discussed. [ABSTRACT FROM AUTHOR]

By introducing shift relations satisfied by a matrix , we propose a generalized Cauchy matrix scheme and construct a discrete second-order Ablowitz–Kaup–Newell–Segur equation. A modified form of this equation is given. By applying an appropriate skew continuum limit, we obtain the semi-discrete counterparts of these two discrete equations; in the full continuum limit, we derive continuous nonlinear equations. Solutions, including soliton solutions, Jordan-block solutions, and mixed solutions, of the resulting discrete, semi-discrete, and continuous Ablowitz–Kaup–Newell–Segur-type equations are presented. The reductions to discrete, semi-discrete, and continuous nonlinear Schrödinger equations and modified nonlinear Schrödinger equation are also discussed. [ABSTRACT FROM AUTHOR]

Some theorems concerning the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation and the characteristics of which are compactly supported and satisfy certain constraints of the theoretical-set type, have been proved. As a consequence, we obtained results on the compact classes of solutions of the corresponding Dirichlet problems considered in a certain Jordan domain. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained. [ABSTRACT FROM AUTHOR]