Behavior Of Solutions Research Articles

Terpyridyl-Imidazole Based Ligand Coordinated to Ln(Hexafluoroacetyl acetonate)3 Core: Synthesis, Structural Characterization, Luminescence Properties, and Thermosensing Behaviors in Solution and PMMA Film.

A new array of ternary lanthanide complexes of the form [Ln(hfa)3(tpy-HImzphen)] have been synthesized and thoroughly characterized wherein Ln = LaIII (1), EuIII (2), SmIII (3), and TbIII (4); hfa = hexafluoroacetylacetonate; and tpy-HImzphen = 2-(4-[2,2':6',2″]terpyridin-4'-yl-phenyl)-1H-phenanthro[9,10-d]imidazole. Incorporation of tpy-HImzphen onto the Ln-hfa moiety induced a bathochromic shift of the absorption window of the complexes into the visible region. Extensive investigations of the luminescence characteristics have been conducted both at RT and at 77 K to understand the deactivation pathways of the complexes. Both steady-state and time-resolved emission spectral behaviors indicate four distinctive behaviors upon incorporation of tpy-HImzphen onto the lanthanide core, viz., a huge red-shift of the ligand-centered peak for LaIII; almost complete energy transfer for EuIII; very little energy transfer for SmIII, while reverse energy transfer in the case of TbIII. In addition, the EuIII-complex exhibits its excellence in luminescence thermometry in the solution state as well as in poly(methyl methacrylate) (PMMA) thin films. The thermosensitive luminescence response in solution was further utilized to mimic set-reset flip-flop logic operation. A plausible energy transfer scheme has been devised to explain dissimilar luminescence behaviors in the complexes. The role of LMCT was also considered for the observed thermosensing property of the Eu(III) complex.

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

This paper introduces gradient-based adaptive neural networks to solve local fractional elliptic partial differential equations. The impact of physics-informed neural networks helps to approximate elliptic partial differential equations governed by the physical process. The proposed technique employs learning the behaviour of complex systems based on input-output data, and automatic differentiation ensures accurate computation of gradient. The method computes the singularity-embedded local fractional partial derivative model on a Hausdorff metric, which otherwise halts the computation by available approximating numerical methods. This is possible because the new network is capable of updating the weight associated with loss terms depending on the solution domain and requirement of solution behaviour. The semi-positive definite character of the neural tangent kernel achieves the convergence of gradient-based adaptive neural networks. The importance of hyperparameters, namely the number of neurons and the learning rate, is shown by considering a stationary anomalous diffusion-convection model on a rectangular domain. The proposed method showcases the network’s ability to approximate solutions of various local fractional elliptic partial differential equations with varying fractal parameters.

Holographic RG flows and boundary conditions in a 3D gauged supergravity

In this work we focus on the study of RG flows of conformal field theories that are holographically dual to Poincaré domain wall solutions in D=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=3$$\\end{document}, N=(2,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}=(2,0)$$\\end{document} gauged supergravity coupled to a sigma model with target space SU(1,1)/U(1)=H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SU}(1, 1)/\ extrm{U}(1) = \\mathbb {H}^2$$\\end{document}. This theory is truncated to a subsector where the vector field and phase of the scalar field vanish and we consider different boundary conditions for the remaining real scalar field. The RG flows, which are mostly non-superysymmetric, are analyzed by treating the supergravity field equations as a dynamical system for the scalar field and its derivative with respect to the scale factor. Phase diagrams are constructed for different values of the parameter a2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a^2$$\\end{document}, which is related to the curvature of the scalar manifold. The behavior of solutions near the boundary is used to determine their type based on the expansion of the corresponding fake superpotential. By incorporating information on the boundary conditions, the obtained RG flows are interpreted using the holographic dictionary. Numerical solutions and plots of the fake superpotential are also provided.

Kamenev-Type Criteria for Testing the Asymptotic Behavior of Solutions of Third-Order Quasi-Linear Neutral Differential Equations

This paper aims to study the asymptotic properties of nonoscillatory solutions (eventually positive or negative) of a class of third-order canonical neutral differential equations. We use Riccati substitution to reduce the order of the considered equation, and then we use the Philos function class to obtain new criteria of the Kamenev type, which guarantees that all nonoscillatory solutions converge to zero. This approach is characterized by the possibility of applying its conditions to a wider area of equations. This is not the only aspect that distinguishes our results; we also use improved relationships between the solution and the corresponding function, which in turn is reflected in a direct improvement of the criteria. The findings in this article extend and generalize previous findings in the literature and also improve some of these findings.

Simulating the Lyotropic Phase Behavior of a Partially Self-Complementary DNA Tetramer.

DNA oligomers in solution have been found to develop liquid crystal phases via a hierarchical process that involves Watson-Crick base pairing, supramolecular assembly into columns of duplexes, and long-range ordering. The multiscale nature of this phenomenon makes it difficult to quantitatively describe and assess the importance of the various contributions, particularly for very short strands. We performed molecular dynamics simulations based on the coarse-grained oxDNA model, aiming to depict all of the assembly processes involved and the phase behavior of solutions of the DNA GCCG tetramers. We find good quantitative matching to experimental data at both levels of molecular association (thermal melting) and collective ordering (phase diagram). We characterize the isotropic state and the low-density nematic and high-density columnar liquid crystal phases in terms of molecular order, size of aggregates, and structure, together with their effects on diffusivity processes. We observe a cooperative aggregation mechanism in which the formation of dimers is less thermodynamically favored than the formation of longer aggregates.

Tri and hexaosmium clusters bearing saccharinate and diphosphine ligands: Synthesis, structure and fluxionality

The reactivity of saccharinate-bridged triosmium cluster [Os3(CO)10(μ-H)(μ-sac)] (1) with two diphosphines namely bis(diphenylphosphino)methane (dppm) and 1,1′-bis(diphenylphosphino)ferrocene (dppf) have been investigated. Cluster 1 reacts with dppm in refluxing toluene to yield triosmium [Os3(CO)9(κ1-dppm)(μ-H)(μ-sac)] (2), [Os3(CO)9(κ1-dppm)(μ-H)(μ-sac)] (3) and [Os3(CO)8(µ-dppm)(μ-H)(µ-sac)] (4) along with the hexaosmium [{Os3(CO)9(μ-H)(µ-sac)}2(κ2-dppm)] (5). The dppm ligand acts as a monodentate ligand in 2 and 3 by coordinating to a single osmium, while it bridges an osmium-osmium edge in 4 and connects to triosmium units in 6. A similar reaction with dppf furnishes [Os3(CO)9(κ1-dppf)(μ-H)(μ-sac)] (6), [Os3(CO)8(µ-dppf)(μ-H)(μ-sac)] (7) and [{Os3(CO)9(μ-H)(µ-sac)}2(κ2-dppf)] (8) which are dppf analogues of 3–5, respectively. Control experiments confirms that 3 is the precursor of 4 and 5, while 6 is the precursor of 7 and 8. All the products are formed via substitution of carbonyl by the diphosphine, and have been characterized by analytical and spectroscopic data. The molecular structures of 2 and 4–7 have been determined by X-ray crystallography. Cluster 7 shows fluxional behavior in solution at room temperature, and its fluxionality has been probed by VT NMR experiments.

The impact of Brownian motion on the optical solutions of the stochastic ultra-short pulses mathematical model

The stochastic generalized nonlinear Schrödinger equation (SGNLSE) of third order in the Stratonovich sense is examined here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are obtained using a modified mapping method. Because the GNLSE is extensively used in nonlinear optical phenomena, optical fiber communication systems, communication and heat pulse propagation in materials, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3-D and 2-D graphs. We infer that multiplicative noise impacts and stabilizes the behavior of SGNLSE solutions.

Structure and Dynamic Rearrangements of the Pt2dba3 and Pd2dba3 Complexes in Solution.

Although the tris(dibenzylideneacetone)diplatinum complex (Pt2dba3) is an important source of Pt(0) used in catalysis and materials science, its structure has not yet been fully elucidated. A thorough study of the three-dimensional structure of Pt2dba3 and its dynamic behavior in solution was carried out using NMR spectroscopy methods at a high field (600 MHz) and molecular modeling. The complex was shown to contain three dba ligands in the s-cis,s-trans, s-trans,s-cis, and s-trans,s-trans conformations, which are uniformly oriented around the Pt2 backbone. In solution, the Pt2dba3 and Pd2dba3 complexes undergo rapid dynamic rearrangements, as evidenced by the exchange between the signals of the olefin protons of various dba ligands in the EXSY NMR spectra. According to the experimental measurements, the activation energies of the rearrangements were estimated to be 19.9 ± 0.2 and 17.9 ± 0.2 kcal/mol for the platinum and palladium complexes, respectively. Three possible mechanisms for this chemical exchange process were considered within the framework of DFT calculations. According to the calculated data, M2dba3 complexes undergo fluxional isomerization involving successive rotations of the dihedral angles formed by the carbonyl group and the C═C bond. Dissociation of dba ligands does not occur within these processes.

Vibrational Self-Consistent Field (VSCF) and Post-VSCF Method Calculations Combined with the Reference Interaction Site Model Self-Consistent Field Method Coupled with the Constrained Spatial Electron Density Distribution: Applications to NaHCOO in Aqueous Phase.

Investigating vibrational behavior in solution is crucial for understanding molecular dynamics within a solvent environment. Notably, the analysis of Raman spectra for molecules in solution is important owing to its ability to unveil intricate solute-solvent interactions. Previous studies have effectively employed frequency calculations utilizing the reference interaction site model self-consistent field method in conjunction with constrained spatial electron density distribution (RISM-SCF-cSED) to understand molecular vibrations in solution, primarily focusing on fundamental vibrational modes. However, the oversight of overtones and combination tones in these studies prompted us to combine the vibrational self-consistent field (VSCF) and vibrational second-order Mo̷ller-Plesset perturbation (VMP2) methods with RISM-SCF-cSED to address these aspects theoretically. Illustrating the efficacy of this integrated approach, we computed the Raman spectra of sodium formate (NaHCOO) in water, revealing the necessity of accounting for molecular anharmonicity in solution vibrational analysis. Our findings underscore the potency of VSCF and VMP2 in conjunction with RISM-SCF-cSED as a robust theoretical framework for such calculations.

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra–Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schläfli integral of the Volterra–Bessel function. In this respect, we study the asymptotic analysis of the Volterra–Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

Synthesis by design and application to desalination of VO2/CTAB-Ti3C2 capacitive deionization electrode

As an emerging technology in capacitive deionization (CDI), flow-electrode capacitive deionization (FCDI) has attracted much attention due to its unlimited electrosorption capacity and continuous operation. Like CDI, carbon is generally chosen as the electrode material for FCDI at this stage, but conventional carbon materials have limited desalination capacity due to the double electric layer. The two-dimensional layered material MXene has been gradually applied to CDI due to its unique structure and physicochemical properties. However, the van der Waals forces exist between the MXene layers, and the layered structure of MXene is susceptible to stacking, while exhibiting an obvious swelling behavior in aqueous solution, thus affecting its desalination performance. Herein, we report a designed VO2/CTAB-Ti3C2 heterostructure synthesised by a simple hydrothermal method. The addition of VO2 can alleviate the lamellar structure buildup of Ti3C2 and improve the desalination performance of Ti3C2. On the other hand, Ti3C2 can be used as a conductive substrate for VO2, which improves the charge transfer rate of VO2, prevents VO2 agglomeration, and provides more contact surfaces for ion intercalation. Compared with the pure VO2 and Ti3C2 electrodes, the composite electrode has a good desalination performance, and its desalination reaches 1041.5 mg g−1 at 1.2 V. Therefore, this VO2/CTAB-Ti3C2 hybrid material is expected to be used as an electrode material for flow capacitive deionization.

Nonlinear Asymptotic Stability and Transition Threshold for 2D Taylor–Couette Flows in Sobolev Spaces

In this paper, we investigate the stability of the 2-dimensional (2D) Taylor–Couette (TC) flow for the incompressible Navier–Stokes equations. The explicit form of velocity for 2D TC flow is given by u=(Ar+Br)(-sinθ,cosθ)T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u=(Ar+\\frac{B}{r})(-\\sin \ heta , \\cos \ heta )^T$$\\end{document} with (r,θ)∈[1,R]×S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(r, \ heta )\\in [1, R]\ imes \\mathbb {S}^1$$\\end{document} being an annulus and A, B being constants. Here, A, B encode the rotational effect and R is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor–Couette flow is asymptotically stable, even at high Reynolds number (Re∼ν-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re\\sim \ u ^{-1}$$\\end{document}), with a sharp exponential decay rate of exp(-ν13|B|23R-2t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\ u ^{\\frac{1}{3}}|B|^{\\frac{2}{3}}R^{-2}t)$$\\end{document} as long as the initial perturbation is less than or equal to ν12|B|12R-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u ^\\frac{1}{2} |B|^{\\frac{1}{2}}R^{-2}$$\\end{document} in Sobolev space. The powers of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} and B in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor–Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient B. The larger the |B|, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor–Couette flows.

Discrete-time staged progression epidemic models

ABSTRACT In the Staged Progression (SP) epidemic models, infected individuals are classified into a suitable number of states. The goal of these models is to describe as closely as possible the effect of differences in infectiousness exhibited by individuals going through the different stages. The main objective of this work is to study, from the methodological point of view, the behaviour of solutions of the discrete time SP models without reinfection and with a general incidence function. Besides calculating R 0 , we find bounds for the epidemic final size, characterize the asymptotic behaviour of the infected classes, give results about the final monotonicity of the infected classes, and obtain results regarding the initial dynamics of the prevalence of the disease. Moreover, we incorporate into the model the probability distribution of the number of contacts in order to make the model amenable to study its effect in the dynamics of the disease.

Effects of Nanoconfinement on Dynamics in Concentrated Aqueous Magnesium Chloride Solutions.

Water behavior in various natural and manufactured settings is influenced by confinement in organic or inorganic frameworks and the presence of solutes. Here, the effects on dynamics from both confinement and the addition of solutes are examined. Specifically, water and ion dynamics in concentrated (2.5-4.2 m) aqueous magnesium chloride solutions confined in mesoporous silica (2.8 nm pore diameter) were investigated using polarization selective pump-probe and 2D infrared spectroscopies. Fitting the rotational and spectral diffusion dynamics measured by the vibrational probe, selenocyanate, with a previously developed two-state model revealed distinct behaviors at the interior of the silica pores (core state) and near the wall of the confining framework (shell state). The shell dynamics are noticeably slower than the bulk, or core, dynamics. The concentration-dependent slowing of the dynamics aligns with behavior in the bulk solutions, but the spectrally separated water-associated and Mg2+-associated forms of the selenocyanate probe exhibit different responses to confinement. The disparity in the complete reorientation times is larger upon confinement, but the spectral diffusion dynamics become more similar near the silica surface. The length scales that characterize the transition from surface-influenced to bulk-like behavior for the salt solutions in the pores are discussed and compared to those of pure water and an organic solvent confined in the same pores. These comparisons offer insights into how confinement modulates the properties of different liquids.

Existence, uniqueness, and asymptotic behaviors of ground state solutions of Kirchhoff‐type equation with fourth‐order dispersion

In this paper, we focus on the following Schrödinger–Kirchhoff‐type problem with fourth‐order dispersion: where are constants and . We make use of Nehari manifold technique together with concentration‐compactness principle to prove that the above equation has at least a ground state solution for if , 6, and 7, and for if . Moreover, we also investigate the asymptotic behaviors of ground state solutions when some coefficients tend to zero. Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow‐up result is established by Pohozaev identity if dimension .

The Schwarzian Approach in Sturm–Liouville Problems

A novel method for finding the eigenvalues of a Sturm–Liouville problem is developed. Following the minimalist approach, the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although the resulting equation is nonlinear, its form allows us to find the general solution by adding a second part to a particular solution. This splitting of the general solution into two parts involves the Schwarzian derivative: hence, the name of the approach. The eigenvalues that correspond to acceptable solutions can be found by requiring the second part to correct the asymptotically diverging behavior of the particular solution. The method can be applied to many different areas of physics, such as the Schrödinger equation in quantum mechanics and stability problems in fluid dynamics. Examples are presented.

Asymptotic behaviour for solutions to reaction–diffusion equations on a root-like metric graph *

In this paper we consider symmetric solutions of reaction–diffusion equations on an unbounded root-like metric graph. We first prove a version of the zero number diminishing properties on the graph, and then use them to show a convergence result for bounded and symmetric solution of general equations. As application, we provide a complete spreading-transition-vanishing trichotomy result on the asymptotic behaviour for solutions of the bistable reaction–diffusion equation on the graph.

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

Influence of organic acids on the viscosity and rheological behavior of guar gum solution

The present study aims to determine the influence of four edible organic acids (ascorbic, citric, malic, and tartaric) at two concentrations (0.5, and 1 %) on the viscosity and rheological behavior of guar gum dispersion (0.2 %, w/v). The finding of this study revealed that the apparent viscosity of guar gum dispersions reduced when the shear rate increased. Additionally, the apparent viscosity of the guar gum dispersions reduced as the organic acids concentration increased from 0 to 1 %. The highest decrease in viscosity was related to 1 % ascorbic acid and the lowest was related to 0.5 % malic acid. The rheological behavior of dispersions were successfully modeled using Power law, Bingham, Herschel-Bulkley, and Casson models, and the Power law model was the best one for describing the behavior of guar gum dispersions containing organic acids. The Power law model showed good performance with the maximum r-value (mean = 0.9988) and least sum of squared error (SSE) values (mean = 0.0021) and root mean square error (RMSE) values (mean = 0.0119) for all samples. The consistency coefficient (k-value) values of the samples (Power law and Herschel-Bulkley models) reduced as the acid percent was increased. The sample containing 1 % tartaric acid had the lowest k-value and the sample containing 0.5 % ascorbic acid had the highest k-value. The flow behavior index (n-value) (Power law and Herschel-Bulkley models) of the samples increased when the acid percent was increased. In summary, the use of guar gum in food products containing high concentrations of tartaric and ascorbic acids is not recommended.

An analysis of stability on the L2‐critical inhomogeneous nonlinear Schrödinger equation

We study the inhomogeneous nonlinear Schrödinger equation in the ‐critical case ( ): We establish a detailed analysis of solutions as the initial data approaches the unstable ground state , which satisfies the stationary equation . The results provide information about singularity formation of the blow‐up solutions and about large‐time asymptotic behavior of the global solutions.