Double Dispersion Equation Research Articles

Global existence and optimal time-decay estimates of solutions to the generalized double dispersion equation on the framework of Besov spaces

We investigate the initial value problem for the generalized double dispersion equation in any dimensions. Inspired by [28] for the hyperbolic system of first order PDEs, we develop Littlewood-Paley pointwise energy estimates for the dissipative wave equation of high-order. Furthermore, with aid of the frequency-localization Duhamel principle, we establish the global existence and optimal decay estimates of solutions in spatially critical Besov spaces. Our results could hold true for any dimensions (n≥1). Indeed, the proofs are different in case of high dimensions and low dimensions owing to interpolation tricks.

Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 + 1)-Dimensional Double Dispersion Equation

This paper considers a generalized double dispersion equation depending on a nonlinear function f ( u ) and four arbitrary parameters. This equation describes nonlinear dispersive waves in 2 + 1 dimensions and admits a Lagrangian formulation when it is expressed in terms of a potential variable. In this case, the associated Hamiltonian structure is obtained. We classify all of the Lie symmetries (point and contact) and present the corresponding symmetry transformation groups. Finally, we derive the conservation laws from those symmetries that are variational, and we discuss the physical meaning of the corresponding conserved quantities.

Invariant preserving schemes for double dispersion equations

We propose and study two finite difference schemes (FDSs) for the double dispersion equations. The first FDS is symplectic, while the second one preserves the discrete momentum exactly. Both FDS conserve the discrete energy approximately with O(h^{2}+tau ^{2}) global error. The extensive numerical experiments agree well with the theoretical results for single solitary wave as well as for the interaction between two solitary waves.

Double dispersion equation for nonlinear waves in a graphene-type hexagonal lattice

It is shown that plane longitudinal nonlinear strain waves in a 2D graphene-type hexagonal lattice are described by a nonlinear double dispersion equation previously developed for the description of waves in an elastic rod. A procedure is developed to derive the governing equation as a continuum limit of the original lattice model. The lattice is described by an interaction of two sub-lattices and both translational and angular interactions between the lattice masses are taken into account.

Jacobi elliptic function solutions of the double dispersive equation in the Murnaghan's rod

This research communication aims to obtain the periodic wave solutions of the double dispersive equation of the wave propagation in a nonlinear elastic inhomogeneous Murnaghan's rod by using the F -expansion technique. This method first converts the partial differential equation into an ordinary differential equation under the wave transformation, then the assumed solution converts the problem under study into systems of algebraic equations. Once these algebraic systems are solved for the unknowns and are shifted into the assumed solution, the exact solutions of the double dispersive equation is obtained. Next by making the modulus of Jacobi elliptic functions into either 0 (or) 1, non-topological, singular and their compound solitons are gleaned. The two- and three-dimensional plots are given to show the roving properties of the solutions.

Blow-Up Phenomena for a Class of Generalized Double Dispersion Equations

In this article, we study the blow-up phenomena of generalized double dispersion equations utt − uxx − uxxt + uxxxx − uxxtt = f(ux)x. Under suitable conditions on the initial data, we first establish a blow-up result for the solutions with arbitrary high initial energy, and give some upper bounds for blow-up time T* depending on sign and size of initial energy E(0). Furthermore, a lower bound for blow-up time T* is determined by means of a differential inequality argument when blow-up occurs.

Existence and asymptotic stability of time periodic solutions to the generalized double dispersion equation with periodic external force

In this paper, we investigate the generalized double dispersion equation with periodic external force in Rn. We prove the existence and uniqueness of time periodic solutions that have the same period as the external force in some suitable function space for the space dimension n≥3. The proof is based on the spectral analysis for the solution operator and the contraction mapping theorem. In addition, we also discuss the time asymptotic stability of the time periodic solutions by continuous argument.

Asymptotic profile of global solutions to the generalized double dispersion equation via the nonlinear term

In this paper, we investigate the initial value problem for the generalized double dispersion equation in $${\mathbb {R}}^n$$ . Weighted decay estimate and asymptotic profile of global solutions are established for $$n\ge 3 $$ . The global existence result was already proved by Kawashima and the first author in Kawashima and Wang (Anal Appl 13:233–254, 2015). Here, we show that the nonlinear term plays an important role in this asymptotic profile.

The initial-boundary value problem for the generalized double dispersion equation

We consider the initial-boundary value problem for the generalized double dispersion equation in all space dimension. Under the suitable assumptions on the initial data and the parameters in the equation, we establish several results concerning local existence, global existence, uniqueness, and finite time blowup property. The exponential decay rate of the energy is proved for global solutions. The sufficient and necessary conditions of global solutions and finite time blowup of solutions are given, respectively.

Asymptotic profile of solutions to the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$

In this paper, we investigate the initial boundary value problem for the linearized double dispersion equation on the half space \begin{document}$\mathbb{R}^{n}_{+}$\end{document} . We convert the initial boundary value problem into the initial value problem by odd reflection. The asymptotic profile of solutions to the initial boundary value problem is derived by establishing the asymptotic profile of solutions to the initial value problem. More precisely, the asymptotic profile of solutions is associated with the convolution of the partial derivative of the fundamental solutions of heat equation and the fundamental solutions of free wave equation.

Lie Symmetry Reductions and Exact Solutions of a Multidimensional Double Dispersion Equation

In this paper, based on classical Lie group method, we study a multidimensional double dispersion equation, and get its infinitesimal generator, symmetry group and similarity reductions. In particular, similarity solutions and travelling wave solutions of the multidimensional double dispersion equation are derived from the reduction equations.

Pointwise estimates of global small solutions to the generalized double dispersion equation

In this paper, we investigate the initial value problem for the generalized double dispersion equation in Rn. Under suitable conditions, global classical solutions are proved for n≥3 and n≥1, respectively. Moreover, pointwise estimates of global classical solutions are established when n≥3 and n is an odd integer by Fourier splitting frequency technique.

Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity

The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed.Grillakis, Shatah and Strauss’ stability theory is applied for the investigation of the orbital stability or instability of solitary waves to the double dispersion equation. An analytical formula, related to some conservation laws of the problem, is derived. As a consequence, the dependence of orbital stability or instability on the parameters of the problem is demonstrated. A complete characterization of the values of the wave velocity, for which the solitary waves to the generalized Boussinesq equation are orbitally stable or unstable, is given.In the special case of a quadratic nonlinearity our results are reduced to those known in the literature.

Stability of Solitary Waves in Biomembranes and Nerves

The orbital stability or instability of solitary waves to the double dispersion equation, modeling the pules propagation in biomembranes and nerves, is investigated. An analytical formula, determining the regions of stability or instability of solitary waves, is derived. Solitary waves with high velocity are treated. The dependence of the regions of stability on the velocity and the parameters of the model is studied.

Asymptotic profile of solutions to the double dispersion equation

In this paper, we investigate the initial value problem for the double dispersion equation in Rn. Decay estimate and asymptotic profile of solutions to the corresponding linear problem are established. The asymptotic profile of solutions to the corresponding linear problem is given by the convolution of the fundamental solutions of heat and free wave equations. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear problem obtained in this paper is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to the nonlinear problem are also established.

Instability and stability properties of traveling waves for the double dispersion equation

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation utt−uxx+auxxxx−buxxtt=−(|u|p−1u)xx for p>1, a>b>0. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms uxxxx and uxxtt. We obtain an explicit condition in terms of a, b and p on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation (b=0), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide analytical as well as numerical results on the variation of the stability region of wave velocities with a, b and p and then state explicitly the conditions under which the traveling waves are orbitally stable.

Solitary waves in auxetic rods with quadratic nonlinearity: Exact analytical solutions and numerical simulations

Special soliton analytical solutions are seeked for the so‐called double dispersion equation (or Porubov's equation), which describes the propagation of longitudinal waves in an elastic rod. Using the F‐expansion method, a new interesting class of traveling solitary waves is obtained. As a byproduct, the results obtained by other authors for some special cases are regained. Some numerical simulations are also performed. Splitting of various initial pulses during propagation into a sequence of solitary waves is considered. The dependence of the amplitude and the velocity of the solitary waves on Poisson's ratio is discussed in detail. Collisions between some obtained solitary waves are also presented. The simulation results are compared with the obtained exact analytical solutions—the latter reflect the perfect balance between the nonlinearity and dispersion. The comparison indicates that the obtained exact solutions are useful for describing more complicated wave fields studied by numerical simulations. It also indicates that the numerical results for auxetic materials are in better agreement with theoretical solutions than in the case of ordinary materials. Furthermore, one can conclude that the Secant pulse generates solitary waves closer to the analytical predictions than a Gaussian pulse. This is similar to the case of variational method, where the Secant trial function is also more proper than the Gaussian one.

Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation

In this paper, we study the initial value problem for the generalized cubic double dispersion equation in n-dimensional space. Under a small condition on the initial data, we prove the global existence and asymptotic decay of solutions for all space dimensions n ≥ 1. Moreover, when n ≥ 2, we show that the solution can be approximated by the linear solution as time tends to infinity.

Global attractor for the generalized double dispersion equation

The paper studies the existence of global attractor for the generalized double dispersion equation arising in elastic waveguide model utt−Δu−Δutt+Δ2u−Δut−Δg(u)=f(x). The main result is concerned with nonlinearities g(u) with supercritical growth. In that case we construct a subclass G of the limit solutions and show that it has a weak global attractor. Especially, in non-supercritical case, the weak topology becomes strong, the further regularity of the global attractor is obtained and the exponential attractor is established in natural energy space.

Some remarks on the stability and instability properties of solitary waves for the double dispersion equation

In this article we give a review of our recent results on the instability and stability properties of travelling wave solutions of the double dispersion equation u(tt) - u(xx) + au(xxxx) - bu(xxtt) = (vertical bar u vertical bar(p-1) u)(xx) for p > 1, a >= b > 0. After a brief reminder of the general class of nonlocal wave equations to which the double dispersion equation belongs, we summarize our findings for both the existence and orbital stability/instability of travelling wave solutions to the general class of nonlocal wave equations. We then state (i) the conditions under which travelling wave solutions of the double dispersion equation are unstable by blow-up and (ii) the conditions under which the travelling waves are orbitally stable. We plot the instability/stability regions in the plane defined by wave velocity and the quotient b/a for various values of p.