We consider long-range variants of Fermi–Pasta–Ulam–Tsingou lattice and, in particular, allow for particles to interact over arbitrarily long distances. We develop sufficient conditions that allow for the construction of solitary wave solutions.
We consider long-range variants of Fermi–Pasta–Ulam–Tsingou lattice and, in particular, allow for particles to interact over arbitrarily long distances. We develop sufficient conditions that allow for the construction of solitary wave solutions.
In this paper, we study the stationary solutions of the inflow problem for a compressible model of the viscous ions motion, which is given by the one-dimensional isentropic compressible Navier–Stokes–Poisson equations. The unique existence of the stationary solutions to the one-dimensional isentropic compressible Navier–Stokes–Poisson equations in the half line is shown provided that the boundary data satisfy some smallness conditions. Moreover, the spatial decay rates of the stationary solutions are presented. By taking the accurate analysis of the cubic characteristic equation for the linearized stationary system, the sign of the real part corresponding to the eigenvalues can be sure. Then our results can be proven by the manifold theory and the center manifold theorem.
Motivated by previous results in special cases associated with Ricci flows, all possible two-components evolutions systems of (1+2)-dimensional second-order partial differential equations (PDEs) admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalization of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible radially symmetric stationary solutions of the Ricci-flow-associated special case are then constructed using the surprisingly rich Lie algebra of the resulting reduced system of ordinary differential equations (ODEs), exemplifying the exceptional status of such systems. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified. By their nature, geometric PDEs typically enjoy rich symmetry properties; our analysis illustrates how those properties may be extrapolated to broader classes of models that are of independent interest.
We investigate the spatiotemporal dynamics of a single-species diffusive model incorporating maturation delay in closed advective heterogeneous environments. First, we establish the well-posedness of the system and prove the existence and uniqueness of the nonconstant positive steady state. Subsequently, we analyze the local stability of the unique nonconstant positive steady state and demonstrate the occurrence of Hopf bifurcation through the corresponding eigenvalue problem. By utilizing a weighted inner product parameterized by the advection rate, we further characterize the stability and direction of the Hopf bifurcation. Finally, we examine how advection rate and spatial length influence the first Hopf bifurcation value, revealing their effects on system dynamics. Our results demonstrate that both advection and spatial scale can either enhance or suppress the likelihood of Hopf bifurcation, depending on the spatial heterogeneity of the intrinsic growth rate.
In nonisothermal setting, microstructural interactions may determine finite-speed heat propagation. We consider such an effect in the dynamics of a viscous incompressible complex fluid (i.e., one with “active” microstructure) through a porous medium. Nonlocal actions and nonlinear damping are considered as determined by the solid–fluid and microstructural interactions. After a choice of constitutive structures and the introduction of a specific truncation of some higher-order terms, we prove existence and uniqueness of global strong solutions to the balance equations. We also analyze pertinent weak solutions.
We consider reaction–diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, colored in space, and invariant under translations. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on timescales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase-tracking mechanism that can be maintained over such long timescales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.
Experiments have shown that mechanical cues play a central role in determining the direction and rate of axonal growth. In particular, neurons seeded on planar substrates undergoing periodic stretching have been shown to reorient and reach a stable equilibrium orientation corresponding to angles within the interval with respect to the main stretching direction. In this work, we present a new model that considers both the reorientation and growth of neurons in response to cyclic stretching. Specifically, a linear viscoelastic model for the growth cone reorientation with the addition of a stochastic term is merged with a moving-boundary model for tubulin-driven neurite growth to simulate the axonal pathfinding process. Various combinations of stretching frequencies and strain amplitudes have been tested by numerical simulation of the proposed model. The simulations show that neurons tend to reorient toward an equilibrium angle that falls in the experimentally observed range. Moreover, the model captures the relation between the stretching condition and the speed of reorientation. Indeed, numerical results show that neurons tend to reorient faster as the frequency and amplitude of oscillation increase.
In this paper, we consider the Korteweg–de Vries (KdV) equation on the real line with an interface. Using Fokas's unified transform method, the explicit solution formulas for the linear forced KdV equation with an interface are derived. Building on these solution formulas, we establish standard estimates for the linear solution and a bilinear estimate for the nonlinear term in a suitable Sobolev space. Using these estimates and a contraction mapping argument, we prove the local well-posedness for the KdV equation with an interface.
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