NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities （BVIs） was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it. Global and local superlinear/ quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.

MERIT FUNCTION AND GLOBAL ALGORITHMFOR BOX CONSTRAINED VARIATIONALINEQUALITIES

The authors consider optimization methods for box constrained variational inequalities. First, the authors study the KKT-conditions problem based on the original problem. A merit function for the KKT-conditions problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original problem is reformulated as minimization with simple constraints. Then, the authors show that any stationary point of the optimization problem is a solution of the original problem. Finally, a descent algorithm is presented for the optimization problem, and global convergence is shown.

New smooth gap function for box constrained variational inequalities

A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.

A NEW SMOOTHING APPROXIMATION METHOD FOR SOLVING BOX CONSTRAINED VARIATIONAL INEQUALITIES

In this paper, we first give a smoothing approximation function of nonsmooth system based on box constrained variational inequalities and then present a new smoothing approximation algorithm. Under suitable conditions,we show that the method is globally and superlinearly convergent. A few numerical results are also reported in the paper.

Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm

In this paper, the shallow water problem is discussed. By treating the incompressible condition as the constraint, a constrained Hamilton variational principle is presented for the shallow water problem. Based on the constrained Hamilton variational principle, a shallow water equation based on displacement and pressure （SWE-DP） is developed. A hybrid numerical method combining the finite element method for spa- tial discretization and the Zu-class method for time integration is created for the SWE- DP. The correctness of the proposed SWE-DP is verified by numerical comparisons with two existing shallow water equations （SWEs）. The effectiveness of the hybrid numerical method proposed for the SWE-DP is also verified by numerical experiments. Moreover, the numerical experiments demonstrate that the Zu-class method shows excellent perfor- mance with respect to simulating the long time evolution of the shallow water.

Summer Atmospheric Water Cycle under the Transition Influence of the Westerly and Summer Monsoon over the Yarlung Zangbo River Basin in the Southern Tibetan Plateau

This study compares the summer atmospheric water cycle,including moisture sources and consumption,in the upstream,midstream,and downstream regions of the Yarlung Zangbo River Basin in the southern Tibetan Plateau.The evolutions of moisture properties under the influence of the westerly and summer southerly monsoon are examined using 5-yr multi-source measurements and ERA5 reanalysis data.Note that moisture consumption in this study is associated with clouds,precipitation,and diabatic heating.Compared to the midstream and downstream regions,the upstream region has less moisture,clouds,and precipitation,where the moisture is brought by the westerly.In early August,the vertical wet advection over this region becomes enhanced and generates more high clouds and precipitation.The midstream region has moisture carried by the westerly in June and by the southerly monsoon from July to August.The higher vertical wet advection maximum here forms more high clouds,with a precipitation peak in early July.The downstream region is mainly affected by the southerly-driven wet advection.The rich moisture and strong vertical wet advection here produce the most clouds and precipitation among the three regions,with a precipitation peak in late June.The height of the maximum moisture condensation is different between the midstream region(325 hPa)and the other two regions(375 hPa),due to the higher upward motion maximum in the midstream region.The diabatic heating structures show that stratiform clouds dominate the upstream region,stratiform clouds and deep convection co-exist in the midstream region,and deep convection systems characterize the downstream region.

EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHRDINGER SYSTEM

We study the existence and stability of the standing waves of two coupled SchrSdinger equations with potentials ｜x｜bi（bi ∈ R,i = 1, 2）. Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the SchrSdinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.

Global solution for coupled nonlinear Klein-Gordon system

The global solution for a coupled nonlinear Klein-Gordon system in two- dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.

Asymptotic Behavior of Least Energy Solutions for a Fractional Laplacian Eigenvalue Problem on R^(N)

We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P)(-△)^(s)u+V(x)u=μu+am(x)|u|^(4s/N)u,∫_(R^(N))|u|^(2)dx=1,u∈H^(s)(R^(N)),where s∈(0,1),μ∈R,a>0,V(x)and m(x)are L^(∞)(R^(N))functions with N≥2.We prove that there is a threshold a^(*)_(s)>0 such that problem(P)has a least energy solution u_(a)(x)for each a∈(0,a^(*)_(s))and u_(a)blows up,as a↗a^(*)_(s),at some point x_(0)∈R^(N),which makes V(x_(0))be the minimum and m(x_(0))be the maximum.Moreover,the precise blowup rates for u_(a)are obtained under suitable conditions on V(x)and m(x).