We are concerned with the nonlinear Schrödinger equation with an L2 mass constraint on both finite and locally finite graphs and prove that the equation has a normalized solution by employing ...variational methods. We also pay attention to the behaviours of the normalized solution as the mass constraint tends to 0+ or +∞ and give clear descriptions of the limit equations. Finally, we provide some numerical experiments on a finite graph to illustrate our theoretical results.
The state of turbulent, minimal-channel flow is estimated from spatio-temporal sparse observations of the velocity, using both a physics-informed neural network (PINN) and adjoint-variational data ...assimilation (4DVar). The performance of PINN is assessed against the benchmark results from 4DVar. The PINN is efficient to implement, takes advantage of automatic differentiation to evaluate the governing equations, and does not require the development of an adjoint model. In addition, the flow evolution is expressed in terms of the network parameters which have a far smaller dimension than the predicted trajectory in state space or even just the initial condition of the flow. Provided adequate observations, network architecture and training, the PINN can yield satisfactory estimates of the flow field, both for the missing velocity data and the entirely unobserved pressure field. However, accuracy depends on the network architecture, and the dependence is not known a priori. In comparison to 4DVar estimation which becomes progressively more accurate over the observation horizon, the PINN predictions are generally less accurate and maintain the same level of errors throughout the assimilation time window. Another notable distinction is the capacity to accurately forecast the flow evolution: while the 4DVar prediction depart from the true flow state gradually and according to the Lyapunov exponent, the PINN is entirely inaccurate immediately beyond the training time horizon unless re-trained. Most importantly, while 4DVar satisfies the discrete form of the governing equations point-wise to machine precision, in PINN the equations are only satisfied in an L2 sense.
•Under-resolved turbulence data are assimilated using 4DVar and PINNs.•4DVar provides the most accurate predictions.•PINNs are robust to measurement noise.•Resolution of observations has significant impact on accuracy of PINN predictions.•Impact of PINN network structure is evaluated.
In this paper we study the following nonlinear Choquard equation −Δu+u=ln1|x|∗F(u)f(u),inR2,where f∈C1(R,R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic ...approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H1(R2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) 11.
In this paper, we aim to investigate the following class of singularly perturbed elliptic problem{−ε2Δu+|x|ηu=|x|ηf(u)inA,u=0on∂A, where ε>0, η∈R, A={x∈R2N:0<a<|x|<b}, N≥2 and f is a nonlinearity of ...C1 class with supercritical growth. By a reduction argument, we show that there exists a nodal solution uε with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimension N−1 as ε→0. In particular, we establish different concentration phenomena of four peaks when the parameter η>2, η=2 and η<2.
In this paper, we are concerned with the following Schrödinger-Born-Infeld system in R3{−Δu+u+λϕu=f(u),x∈R3,−div(∇ϕ1−|∇ϕ|2)=λu2,x∈R3,u(x)→0,ϕ(x)→0,as|x|→∞, where λ is a positive parameter, f∈C1(R,R) ...is a general nonlinearity introduced by Berestycki and Lions. The existence of nontrivial solutions for small λ and the asymptotic behavior of these solutions as λ tends to zero are established by variational methods.
The Newton method is used for optimisation in the maximum likelihood ensemble filter (MLEF) to improve analysis convergence and accuracy. The proposed method is compared against the original method ...using the conjugate gradient (CG) method preconditioned by the Hessian for optimisation. The mechanisms of the two minimisation methods are illustrated with optimisation for the Booth and Rosenbrock functions. Comparisons are then made in simple data assimilation experiments. In the assimilation of a single wind speed, the Newton method is affected by the gradient and Hessian approximated by the forecast ensemble but the gradient norm decreases geometrically. The CG method is terminated at the first step unless the ensemble perturbation matrix in the observation space is fixed. In the cycled experiments using a Korteweg–de Vries–Burgers equation model with a quadratic observation operator, the Newton method and the preconditioned CG method with gradients updated during iterations yield an analysis with comparable accuracy, but the CG with the fixed gradient is found to produce an analysis that leads to unstable forecast. When the number of Newton iterations is limited to one, the solutions remain suboptimal, significantly destabilising the model. The experimental results indicate that the Newton method is a promising alternative to the CG method with a line search for optimisation in MLEF.
We consider the following k-coupled nonlinear Schrödinger systems:{−Δuj+λjuj=μjuj3+∑i=1,i≠jkβi,jui2ujin RN,uj>0in RN,uj(x)→0as |x|→+∞,j=1,2,⋯,k, where N≤3, k≥3, λj,μj>0 are constants and βi,j=βj,i≠0 ...are parameters. There have been intensive studies for the above systems when k=2 or the systems are purely attractive (βi,j>0,∀i≠j) or purely repulsive (βi,j<0,∀i≠j); however very few results are available for k≥3 when the systems admit mixed couplings and the components are organized into groups, i.e., there exist (i1,j1) and (i2,j2) such that βi1,j1>0 and βi2,j2<0. In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the systems admit mixed couplings and the components are organized into groups. We first divide these systems into repulsive-mixed and total-mixed cases. In the first case we prove nonexistence of ground states. In the second case we give a necessary condition for the existence of ground states and also provide estimates for Morse index. The key idea is the block decomposition of the systems (optimal block decompositions, eventual block decompositions), and the measure of total interaction forces between different blocks. Finally the assumptions on the existence of ground states are shown to be optimal in some special cases.
Nous considérons les systèmes Schrödinger k-couplés non-linéaires ci-dessous :{−Δuj+λjuj=μjuj3+∑i=1,i≠jkβi,jui2ujin RN,uj>0in RN,uj(x)→0as |x|→+∞,j=1,2,⋯,k, où N≤3, k≥3, λj,μj>0 sont les constants et βi,j=βj,i≠0 les paramètres. Pour les systèmes ci-dessus, quand k=2 ou quand ils sont purement attractifs (βi,j>0,∀i≠j) ou répulsifs (βi,j<0,∀i≠j) ; un grand nombre d'études et de recherches sont déjà effectuées. Pourtant, il manque encore des résultats valables sur k≥3 quand les systèmes sont mixtes et couplants et que les composants sont organisés en de différentes parties, soit quand (i1,j1) et (i2,j2) font βi1,j1>0 et βi2,j2<0. Dans le présent article, nous allons prendre l'initiave de faire une étude systématique et quasi complète sur l'existence de états fondamentaux. Nous allons d'abord diviser les systèmes en deux catégories : des systèmes mixtes-replusifs et des systèmes mixtes-totaux. Pour les systèmes mixtes-replusifs, nous avons prouvé leur non-existence. Et pour les systèmes mixtes-totaux, nons avons donné une condition nécesseaire de l'existence de états fondamentaux et nous avons fait également des estimations de l'index Morse. Notre idée-clé consite dans la décomposition des blocs de systèmes (décompositions des blocs optimaux, décompositions des blocs éventuels) et dans le calcul du total des forces interactionnelles entre différents blocs. Enfin, nous avons prouvé que dans certains cas particuliers, nos présomptions données sur l'existence de états fondamentaux sont les plus optimaux.
Consider nonlinear Choquard equations −Δu+u=(Iα∗F(u))F′(u)in RN,limx→∞u(x)=0,where Iα denotes Riesz potential and α∈(0,N). In this paper, we show that when F is doubly critical, i.e. ...F(u)=NN+α|u|N+αN+N−2N+α|u|N+αN−2, the nonlinear Choquard equation admits a nontrivial solution if N≥5 and α+4<N.
•A two-component variational image denoising model is established.•The existence and uniqueness of solution for the variational model are proved.•A numerical algorithm for this model is proposed to ...validate the theoretical resultsl.•The efficiency of our numerical method is tested by comparing with some other published works.
Image denoising is to recover true image from noisy image. Many image deonising models are proposed during the last decades. Some models preserve the margin of tissue, i.e., TV model, while the others, i.e., LLT model, prefer smooth solutions. By decomposing true image into cartoon part and texture part, we propose a fractional image denoising model with two-component regularization terms. Setting some appropriate parameters, the proposed model can deal with both smooth and non-smooth image denosing problems. The existence and uniqueness of solution for the variational model are proved. Moreover, a Split-Bregman(S-B) based numerical algorithm to solve this model is also proposed to validate the theoretical results. Numerical tests show that the proposed model can produce competitive denoising result to the other three published models.
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