On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.     

On the nonlinear self-adjointness of the Zakharov–Kuznetsov equation

A new general theorem, which does not require the existence of Lagrangians, allows to compute conservation laws for an arbitrary differential equation. This theorem is based on the concept of self-adjoint equations for nonlinear equations. In this paper we show that the Zakharov–Kuznetsov equation is self-adjoint and nonlinearly self-adjoint. This property is used to compute conservation laws corresponding to the symmetries of the equation. In particular the property of the Zakharov–Kuznetsov equation to be self-adjoint and nonlinearly self-adjoint allows us to get more conservation laws.     

On Doi–Naganuma and Shimura liftings

The Ramanujan Journal - In this paper, we extend the Doi–Naganuma lifting to higher levels by following the methods of Zagier and Kohnen. We prove that there is a Hecke-equivariant linear map...     

On the high temperature region of the Sherrington–Kirkpatrick model

We prove the validity of the replica-symmetric formulas for the Sherrington–Kirkpatrick (SK) model in a region of parameters that (probably) coincides with the region predicted by the physicists.     

On the ground states of the one-dimensional Falicov–Kimball model

The ground states of the one-dimensional Falicov–Kimball model are studied in the grand canonical ensemble for large values of the interaction strength U. The quantum particle chemical potential μ_e is chosen in the interval −U+4<μ_e<0, such that, then, these states are neutral states and depend only on the sum of the two chemical potentials, μ=μ_i+μ_e. Consequences of this study are, among others, the following results. If ρ=p/q (p and q relatively prime) is a rational number we prove that, for U⩾U₀(q) (where U₀(q) is a specific function), there is an interval on the μ-axis, of length larger than qU^−2q+3, such that for any μ in this interval, the ground state has density ρ. In this interval the ground state is unique, up to translations, and the corresponding classical particle configuration is described by the characteristic sequence associated with the rational number ρ.     

On the asymptotic distribution of the Koenker–Bassett estimator for a parameter of the nonlinear model of regression with strongly dependent noise

We prove that, under certain regularity conditions, the asymptotic distribution of the Koenker – Bassett estimator coincides with the asymptotic distribution of the integral of indicator process generated by a random noise weighted by the gradient of the regression function.     

On nonlinear analogues of the Phragmen–Lindelöf theorem

For solutions of quasilinear elliptic inequalities containing lower-order derivatives, we obtain estimates of the growth that take into account the geometry of the domain. Corollaries of these results are nonlinear analogues of the Phragmen–Lindelöf theorem.     

Pacemaker dynamics in the full Morris–Lecar model

This article reports the finding of pacemaker dynamics in certain region of the parameter space of the three-dimensional version of the Morris–Lecar model for the voltage oscillations of a muscle cell. This means that the cell membrane potential displays sustained oscillations in the absence of an external electrical stimulation. The development of this dynamic behavior is shown to be tied to the strength of the leak current contained in the model. The approach followed is mostly based on the use of linear stability analysis and numerical continuation techniques. In this way it is shown that the oscillatory dynamics is associated to the existence of two Hopf bifurcations, one subcritical and other supercritical. Moreover, it is explained that in the region of parameter values most commonly studied for this model such pacemaker dynamics is not displayed because of the development of two fold bifurcations, with the increase of the strength of the leak current, whose interaction with the Hopf bifurcations destroys the oscillatory dynamics.     

Local solutions of the fast–slow model of an optoelectronic oscillator with delay

Theoretical and Mathematical Physics - We study a difference–differential model of an optoelectronic oscillator that is a modification of the Ikeda equation with delay. We analyze the...     

On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows

This article concerns large time behavior of Ladyzhenskaya model for incompressible viscous flows in R~3.Based on linear L~P-L~q estimates,the auxiliary decay properties of the solutions and generalized Gronwall type arguments,some optimal upper and lower bounds for the decay of higher order derivatives of solutions are derived without assuming any decay properties of solutions and using Fourier splitting technology.     

On the dynamics of the Lengyel–Epstein model with forcing intensity

This paper is concerned with the Lengyel–Epstein model for interacting chemicals under Dirichlet–Neumann boundary data. This model describe the reaction between iodide, malonic and clorite acid (CIMA reaction). In particular the Lengyel–Epstein model that takes into account the effect of illumination of the reaction cell is investigated. It is shown that the solutions are bounded. The linear stability of the steady states is discussed. Conditions guaranteeing the nonlinear stability are also obtained.

     

On the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation

In this note, we establish analytically the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation. The existence and uniqueness of positive, bounded and monotone solutions for this scheme was recently established in [J. Diff. Eq. Appl. 19 (2014), pp. 1907–1920]. In the present work, we prove additionally that the method is convergent of order one in time, and of order two in space. Some numerical experiments are conducted in order to assess the validity of the analytical results. We conclude that the methodology under investigation is a fast, nonlinear, explicit, stable, convergent numerical technique that preserves the positivity, the boundedness and the monotonicity of approximations, making it an ideal tool in the study of some travelling-wave solutions of the mathematical model of interest. This note closes proposing new avenues of future research.     

The Riemann problem for the nonisentropic Baer–Nunziato model of two-phase flows

The Riemann problem for the well-known Baer–Nunziato model of two-phase flows is solved. The system consists of seven partial differential equations with nonconservative terms. The most challenging problem is that this model possesses a double eigenvalue. Although characteristic speeds coincide, the curves of composite waves associated with different characteristic fields can be still constructed. They will also be incorporated into composite wave curves to form solutions of the Riemann problem. Solutions of the Riemann problem will be constructed when initial data are in supersonic regions, subsonic regions, or in both kinds of regions. A unique solution and solutions with resonance are also obtained.     

On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.     

Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system

This paper is concerned with the existence, uniqueness, and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes–Korteweg system effected by the given mass source, the external force of general form, and the energy source in R³${\mathbb{R}}^3$. Based on the weighted L²$L^2$-method and some delicate L^∞$L^{\infty }$ estimates on solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier–Stokes–Korteweg system.     

On the concentration of semi-classical states for a nonlinear Dirac–Klein–Gordon system

In the present paper, we study the semi-classical approximation of a Yukawa-coupled massive Dirac–Klein–Gordon system with some general nonlinear self-coupling. We prove that for a constrained coupling constant there exists a family of ground states of the semi-classical problem, for all ?   small, and show that the family concentrates around the maxima of the nonlinear potential as ?→0$?\to 0$. Our method is variational and relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.     

On the characterization of the nonlinear Schrödinger hierarchy

It is proved that the conserved polynomials of the nonlinear Schrödinger equation have a vanishing residue property analogous to those now known to characterize the Korteweg-de Vries, Modified Korteweg-de Vries and Sine-Gordon hierarchies.     

Spatial patterns of the Holling–Tanner predator–prey model with nonlinear diffusion effects

In this article, we study the Holling–Tanner predator–prey model with nonlinear diffusion terms under homogeneous Neumann boundary condition. The nonlinear diffusion terms here mean that the prey runs away from the predator, and the predator chases the prey. Nonexistence and existence of nonconstant positive steady states are obtained, which reveal that cross-diffusion can create spatial patterns even when the random diffusion fails to do so. Moreover, asymptotic behaviour of positive solutions as the cross-diffusion tends to ∞ is shown.