High-dimensional Equations Research Articles

An efficient Monte Carlo scheme for Zakai equations

In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman–Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of an integrable evolution equation in one spatial dimension. Do there exist integrable analogs of the modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Multiple soliton solutions, lump, rogue wave and breather solutions of high dimensional equation for describing Rossby waves

In the past, we haven't paid much attention to higher-dimensional models, which are actually more consistent with the real atmosphere. In this manuscript, we derive a high-dimensional Kadomtsev-Petviashvili equation from a fluid system based on the Gardner-Morikawa transformation and the small parameter perturbation method, which is the first time to characterize the propagation process of Rossby waves in Large-scale atmospheric. The exact solutions such as soliton, lump, rogue wave and breather of this equation are achieved according to Hirota bilinear method and symbolic computation approach. Further we utilize the composite diagram of these solutions to observe the propagation properties of Rossby waves with assistance of Mathematica. The results enrich the analysis of Rossby waves in marine engineering and fluid dynamics.

New Analytic Solutions for Fluid Flow Equations in Higher Dimensions Around an Offshore Structure Describing Bidirectional Wave Surfaces

New Analytic Solutions for Fluid Flow Equations in Higher Dimensions Around an Offshore Structure Describing Bidirectional Wave Surfaces

Long time behaviour of finite-energy radial solutions to energy subcritical wave equation in higher dimensions

We consider the defocusing, energy subcritical wave equation ∂t2u−Δu=−|u|p−1u in 4 to 6 dimensional spaces with radial initial data. We define w=r(d−1)/2u, reduce the equation above to one-dimensional equation of w and apply method of characteristic lines. This gives scattering of solutions outside any given light cone as long as the energy is finite. The scattering in the whole space can also be proved if we assume the energy decays at a certain rate as x→+∞. This generalizes the 3-dimensional results in Shen [32] to higher dimensions.

A mass‐ and energy‐preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions

AbstractThis article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of with time step and mesh size in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete norm or norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.

Global well-posedness, scattering and blow-up for the generalized Boussinesq equation in high dimensions below the ground state energy

In this paper, we study the generalized Boussinesq equation in energy space. For spatial dimensions d≥3, nonlinear exponent α+1∈[1+4/d,1+4/(d−2)), initial data (u0,u1), we clarify the longtime behavior of the solution with energy E(u0,u1) below the ground state energy E(Qd,α,0). It depends on K(u0)=‖∇u0‖L2dα/2−2‖u0‖L22−(d−2)α/2. For K(u0)>K(Qd,α), the solution blows up in finite time. For K(u0)<K(Qd,α), the solution is global and scatters to a linear solution in energy space if u0,u1 are radial functions. To show the scattering, we use the concentration-compactness argument of Kenig-Merle [15] and the Morawetz-virial type estimate obtained in [7].

Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

Radiating stars and Riccati equations in higher dimensions

The objective of this study is to investigate spherically symmetric radiating stars undergoing gravitational collapse, in higher dimensional general relativity, inclusive of acceleration, expansion, shear, an electromagnetic field and a cosmological constant. Methods that can be used to obtain exact solutions to the boundary condition with/without a linear equation state are studied. Two distinct approaches are investigated. In the first approach, the boundary condition is expressed as a Riccati equation in terms of one of the dependent variables, and restrictions are placed to obtain new exact solutions. In the second approach, transformations that map the boundary condition into a new Riccati equation are investigated. The resulting new transformed equation is solved, by placing restrictions on the coefficients, to obtain new exact models. Special properties of the transformation are shown when appropriate restrictions on the parameters of the transformation are placed. This allows the order of the boundary condition to be reduced from a second order partial differential equation into a first order partial differential equation. The versatility of the transformation on other equations is exhibited when new solutions to the system of equations consisting of both the boundary condition and equation of state are obtained. When the dimension is set to four, some known solutions are recovered. It is shown that horizons can be identified by using a special case of the transformation. Our results elucidates the importance of the use of transformations that map the coordinates of differential equations into new and different coordinate systems.

Improved local smoothing estimate for the wave equation in higher dimensions

In this paper, we establish the sharp k-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions n≥3. As a byproduct, we also generalize the restriction estimates of Ou–Wang [21] to a broader class of phase functions.

Global well-posedness and exponential decay for the inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation

We consider the Cauchy problem for the inhomogeneous incompressible logarithmical hyper-dissipative Navier-Stokes equations in higher dimensions. By means of the Littlewood-Paley techniques and new ideas, we establish the existence and uniqueness of the global strong solution with vacuum over the whole space R n \mathbb {R}^{n} . Moreover, we also obtain the exponential decay-in-time of the strong solution. Our result holds without any smallness on the initial data and the initial density is allowed to have vacuum.

A new ‘walk on spheres’ type method for fractional diffusion equation in high dimensions based on the Feynman–Kac formulas

In this paper, we introduce an efficient stochastic method for solving space-fractional diffusion equations in high dimensions based on the Feynman–Kac formula. The key idea is to approximate the trajectory of the process by using a series of balls. As an extension of our first work Sheng et al. (2022) for the Poisson equation, the new algorithm finds remarkably efficient in solving time-dependent linear problems with integral fractional Laplacian on the bounded and unbounded domains. We present some numerical examples to validate the robustness and efficiency of our methods.

A random batch method for efficient ensemble forecasts of multiscale turbulent systems.

A new efficient ensemble prediction strategy is developed for a multiscale turbulent model framework with emphasis on the nonlinear interactions between large and small-scale variables. The high computational cost in running large ensemble simulations of high-dimensional equations is effectively avoided by adopting a random batch decomposition of the wide spectrum of the fluctuation states, which is a characteristic feature of the multiscale turbulent systems. The time update of each ensemble sample is then only subject to a small portion of the small-scale fluctuation modes in one batch, while the true model dynamics with multiscale coupling is respected by frequent random resampling of the batches at each time updating step. We investigate both theoretical and numerical properties of the proposed method. First, the convergence of statistical errors in the random batch model approximation is shown rigorously independent of the sample size and full dimension of the system. Next, the forecast skill of the computational algorithm is tested on two representative models of turbulent flows exhibiting many key statistical phenomena with a direct link to realistic turbulent systems. The random batch method displays robust performance in capturing a series of crucial statistical features with general interests, including highly non-Gaussian fat-tailed probability distributions and intermittent bursts of instability, while requires a much lower computational cost than the direct ensemble approach. The efficient random batch method also facilitates the development of new strategies in uncertainty quantification and data assimilation for a wide variety of general complex turbulent systems in science and engineering.

Isotropic Perfect Fluids in Modified Gravity

We generate the Einstein–Gauss–Bonnet field equations in higher dimensions for a spherically symmetric static spacetime. The matter distribution is a neutral fluid with isotropic pressure. The condition of isotropic pressure, an Abel differential equation of the second kind, is transformed to a first order nonlinear canonical differential equation. This provides a mechanism to generate exact solutions systematically in higher dimensions. Our solution generating algorithm is a different approach from those considered earlier. We show that a specific choice of one potential leads to a new solution for the second potential for all spacetime dimensions. Several other families of exact solutions to the condition of pressure isotropy are found for all spacetime dimensions. Earlier results are regained from our treatments. The difference with general relativity is highlighted in our study.

Finite time blow up and concentration phenomena for a solution to drift-diffusion equations in higher dimensions

We show the finite time blow up of a solution to the Cauchy problem of a drift-diffusion equation of a parabolic-elliptic type in higher space dimensions. If the initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not exist globally in time and blows up in a finite time for the scaling critical space. Besides there exists a concentration point such that the solution exhibits the concentration in the critical norm. This type of blow up was observed in the scaling critical two dimensions. The proof is based on the profile decomposition and the Shannon inequality in the weighted space.

Applying the natural transform iterative technique for fractional high-dimension equations of acoustic waves

In the present study, the natural transform iterative method (NTIM) has been implemented for the solution of a fractional Zakharavo–Kuznetsov (FZK) equation. NTIM is a relatively new technique for handling non-linear fractional differential equations. The method is tested upon the two non-linear FZK equalities. The solution of the proposed technique has been compared with the existing perturbation–iteration algorithm (PIA) method and residual power series method (RPSM). From the numerical results, it is clear that the method handles non-linear differential equations very suitably and provides the results in very closed accord with the accurate solution. As a result, the NTIM approach is regarded as one of the finest analytical techniques for solving fractional-order linear and non-linear problems.

Normal forms for saddle-node bifurcations: Takens’ coefficient and applications in climate models

We show that a one-dimensional differential equation depending on a parameter μ with a saddle-node bifurcation at μ = 0 can be modelled by an extended normal form y ˙ = ν ( μ ) − y 2 + a ( μ ) y 3 , where the functions ν and a are solutions to equations that can be written down explicitly. The equivalence to the original equations is a local differentiable conjugacy on the basins of attraction and repulsion of stationary points in the parameter region for which these exist, and is a differentiable conjugacy on the whole local interval otherwise. (Recall that in standard approaches local equivalence is topological rather than differentiable.) The value a ( 0 ) is Takens’ coefficient from normal form theory. The results explain the sense in which normal forms extend away from the bifurcation point and provide a new and more detailed characterization of the saddle-node bifurcation. The one-dimensional system can be derived from higher dimensional equations using centre manifold theory. We illustrate this using two examples from climate science and show how the functions ν and a can be determined analytically in some settings and numerically in others.

Time-fractional telegraph equation with ψ-Hilfer derivatives

This paper deals with the investigation of the solution of the time-fractional telegraph equation in higher dimensions with ψ-Hilfer fractional derivatives. By application of the Fourier and ψ-Laplace transforms the solution is derived in closed form in terms of bivariate Mittag-Leffler functions in the Fourier domain and in terms of convolution integrals involving Fox H-functions of two-variables in the space-time domain. A double series representation of the first fundamental solution is deduced for the case of odd dimension. The results derived here are of general nature since our fractional derivatives allow to interpolate between Riemann-Liouville and Caputo fractional derivatives and the use of an arbitrary positive monotone increasing function ψ in the kernel allows to encompass most of the fractional derivatives in the literature. In the one dimensional case, we prove the conditions under which the first fundamental solution of our equation can be interpreted as a spatial probability density function evolving in time, generalizing the results of Orsingher and Beghin (2004). Some plots of the fundamental solutions for different fractional derivatives are presented and analysed, and particular cases are addressed to show the consistency of our results.2010 MSC: 35R11, 26A33, 35A08, 35A22, 35C15, 60G22.

Steady‐state solutions of the aerotaxis problem

We study the steady‐state system of aerotaxis equations in higher dimensions. It is shown that the existence and multiplicity of solutions depend on the total mass of the colony of bacteria, the energy function, and the boundary conditions.

Mixed dispersion nonlinear Schrödinger equation in higher dimensions: theoretical analysis and numerical computations

In the present work we provide a characterization of the ground states of a higher-dimensional quadratic-quartic model of the nonlinear Schrödinger class with a combination of a focusing biharmonic operator with either an isotropic or an anisotropic defocusing Laplacian operator (at the linear level) and power-law nonlinearity. Examining principally the prototypical example of dimension d = 2, we find that instability arises beyond a certain threshold coefficient of the Laplacian between the cubic and quintic cases, while all solutions are stable for powers below the cubic. Above the quintic, and up to a critical nonlinearity exponent p, there exists a progressively narrowing range of stable frequencies. Finally, above the critical p all solutions are unstable. The picture is rather similar in the anisotropic case, with the difference that even before the cubic case, the numerical computations suggest an interval of unstable frequencies. Our analysis generalizes the relevant observations for arbitrary combinations of Laplacian prefactor b and nonlinearity power p.