Saddle Solutions Research Articles

Saddle solutions for the planar Schrödinger–Poisson system with exponential growth

Saddle solutions for the planar Schrödinger–Poisson system with exponential growth

Dynamic Relationship Between KKT Saddle Solutions and Optimal Solutions in AC OPF Problems

Many KKT condition-based methods for solving AC OPF models can compute KKT-saddle solutions that are feasible solutions but not local optimal solutions. This paper explores a dynamic relationship between KKT-saddle solutions and optimal solutions in AC OPF problems and presents a “buy one, get two free” theorem by showing that if one KKT saddle is computed, then this KKT saddle is dynamically connected to two local optimal solutions. In addition, it is shown that by following the unstable manifold of the computed KKT saddle, one can easily compute two local optimal solutions. This “buy one, get two free” property occasionally degenerates into a “buy one, get one free” property when the unstable equilibrium point (UEP) associated with the computed KKT saddle does not lie on a quasi-stability boundary. Numerical demonstration of these two properties is conducted on several test systems including the IEEE 118-bus system. The robustness of these two properties under the perturbations of AC OPF models is analytically and numerically shown.

Revision of the standard EN 1993‐1‐11 and background information

AbstractWithin the framework of the development of the second generation of Eurocode 3, Part 1‐11, related to the design of steel tension components, was completely revised.This paper introduces the main developments in the prEN 1993‐1‐11:2023 regarding the basis of design for the three groups of tension components: A ‐ Rods, B ‐ Ropes, and C ‐ Parallel strand or wire systems. For each of the three groups, complete information on the “Material”, “Ultimate limit state” (including fatigue) and “Serviceability limit state” requirements are provided. Improved formulas for the design of saddles and clamps are presented, including new saddle solutions for group C. Reference is made to the new group D ‐ Parallel wire tension components for suspended structures, included in new Annex C. Annex A “Product requirements for tension components” was completely revised to reflect modern testing requirements. Informative Annex B has been also updated with new figures that reflect the state of the art in tension elements and terminations.

Resurgence in 2-dimensional Yang–Mills and a genus-altering deformation

Abstract We study resurgence in the context of the partition function of 2-dimensional SU(N) and U(N) Yang–Mills theory on a surface of genus h. After discussing the properties of the transseries in the undeformed theory, we add a term to the action to deform the theory. The partition function can still be calculated exactly, and the deformation has the effect of analytically continuing the effective genus parameter in the exact answer so that it is noninteger. In the deformed theory we find new saddle solutions and study their properties. In this context each saddle contributes an asymptotic series to the transseries which can be analyzed using Borel-Écalle resummation. For specific values of the deformation parameter we find Cheshire cat points where the asymptotic series in the transseries truncate to a few terms. We also find new partial differential equations satisfied by the partition function, and a number of applications of these are explained, including low-order/low-order resurgence.

Normalized saddle solutions for a mass supercritical Choquard equation

We investigate the existence of saddle type normalized solutions for the nonlinear Choquard equation:{−Δu−λu=(Iα⁎F(u))F′(u), in RN∫RN|u|2dx=a2,u∈H1(RN). Here N≥1, a>0 is given in advance, Iα is the Riesz potential of order α∈(0,N) and the unknown parameter λ appears as a Lagrange multiplier. In a mass supercritical setting on F, we prove the existence of a couple (uaG,λaG)∈H1(RN)×R− of saddle solutions for any a>0 and for given finite Coxeter group G with its rank k≤N. Our method is to combine the concentration compactness principle with a minimax procedure in the saddle type symmetric subspace, which gives a variational framework of constructing normalized saddle solutions for the Choquard equation.

Hessian estimates for saddle solutions of a special Lagrangian equation in dimension three

We obtain a priori interior Hessian estimates for saddle solutions of the equation Δu=det⁡D2u in dimension 3 in the form of Laplace integral, with almost necessary conditions, verified by our counterexamples. Other sufficient conditions for uniform Hessian estimates are also included.

Revisiting viscous transonic decretion disks of Be stars

Aims. In the context of Be stars, we re-studied the viscous transonic decretion disk model of these stars. This model is driven by a radiative force due to an ensemble of optically thin lines and viscosity considering the Shakura–Sunyaev prescription. Methods. The nonlinear equation of motion presents a singularity (sonic point) and an eigenvalue, which is also the initial condition at the stellar surface. Then, to obtain this eigenvalue, we set it as a radial quantity and performed a detailed topological analysis. We describe a numerical method for solving either nodal or saddle transonic solutions. Results. The value of the viscosity α barely determines the location of the sonic point, but it determines the topology of the solution. We found two nodal solutions, which are almost indistinguishable. Saddle solutions were found for lower values of α than required for the nodal solutions. In addition, rotational velocity does not play a determinate role in the velocity (and density) profile, because the viscosity effects collapse all the solutions to almost a unique one in a small region above the stellar surface. Conclusions. A suitable combination of line-force parameters and/or disk temperature gives the location of the sonic point lower than 50 stellar radii, describing a truncated disk. This could explain the SED turn-down observed in Be stars without needing a binary companion.

Saddle solutions for the Choquard equation with a general nonlinearity

In the spirit of Berestycki and Lions (Arch. Rational Mech. Anal., 82: 313–345, 1983), we prove the existence of saddle-type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u + u= \big (I_\alpha *F(u)\big )F'(u)\qquad \text { in }\;\mathbb {R}^N \end{aligned}$$ where $$N\ge 2$$ and $$I_\alpha$$ is the Riesz potential of order $$\alpha \in (0,N)$$ . Given a finite Coxeter group G with rank $$k\le N$$ , we construct a G-groundstate uniformly with lowest energy amongst G-saddle solutions for the Choquard equation in a noncompact setting. Moreover, if $$F'$$ is odd and has constant sign on $$(0,+\infty )$$ , then every G-groundstate maintains signed on the fundamental domain of the corresponding Coxeter group and receives opposite signs on any two adjacent regions so that nodal domains of G-groundstate are of cone shapes demonstrating Coxeter’s symmetric configurations in $$\mathbb {R}^N$$ . These results further complete the variational framework in constructing sign-changing solutions for the Choquard equation but still require a quadratic or super-quadratic growth on F near the origin.

Traveling pulses with oscillatory tails, figure-eight-like stack of isolas, and dynamics in heterogeneous media

The interplay between 1D traveling pulses with oscillatory tails (TPOs) and heterogeneities of the bump type is studied to formulate a generalized three-component FitzHugh–Nagumo system of equations. We present that stationary pulses with oscillatory tails (SPOs) form a “snaky” structure in homogeneous spaces and then TPO branches form a “figure-eight-like stack of isolas” located adjacent to the snaky structure of the SPO. Here, we adopt input resources such as the voltage-difference as a bifurcation parameter. A drift bifurcation from the SPO to TPO is observed by introducing another parameter at which these two solution sheets merge. In contrast to the monotone tail case, in the heterogeneous problem, a nonlocal interaction appears between the TPO and the heterogeneity, creating infinitely many saddle solutions and finitely many stable stationary solutions distributed across the line. The response of the TPO shows various dynamics including pinning and depinning processes along with penetration (PEN) and rebound (REB). The stable/unstable manifolds of these saddles interact with the TPO in a complex manner, creating a subtle dependence on the initial condition, and a difficulty to predict the behavior after collision even in 1D space. For the 1D case, a systematic global exploration of solution branches induced by heterogeneities (heterogeneity-induced-ordered patterns; HIOPs) reveals that HIOPs contain all the asymptotic states after collision to facilitate the prediction of the solution results without solving the PDEs. The reduction method of finite-dimensional system enables the clarification of the detailed mechanism of the transitions from PEN to pinning and pinning to REB based on a dynamical system perspective. Consequently, the basin boundary between two distinct outputs against the heterogeneities yields infinitely many successive reconnections of heteroclinic orbits among those saddles when the strength of heterogeneity increases. This causes the aforementioned subtle dependence of initial condition.

Saddle solutions for the fractional Choquard equation

We study the saddle solutions for the fractional Choquard equation \begin{align*} (-\Delta)^{s}u+ u=(K_{\alpha}\ast|u|^{p})|u|^{p-2}u, \quad x\in \mathbb{R}^N \end{align*} where $s\in(0,1)$, $N\geq 3$ and $K_\alpha$ is the Riesz potential with order $\alpha\in (0,N)$. For every Coxeter group $G$ with rank $1\leq k\leq N$ and $p\in[2,\frac{N+\alpha}{N-2s})$, we construct a $G$-saddle solution with prescribed symmetric nodal configurations. This is a counterpart for the fractional Choquard equation of saddle solutions to the Choquard equation and further completes the existence of non-radial sign-changing solutions for this doubly nonlocal equation.

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solutionC±with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

Saddle solutions for the critical Choquard equation

We study the saddle type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u = (I_\alpha *|u|^{\frac{N+\alpha }{N-2}} )|u|^{\frac{N+\alpha }{N-2}-2}u \quad \text { in }\;{\mathbb {R}}^N, \end{aligned}$$ where $$N\ge 3$$ , $$I_\alpha $$ is the Riesz potential of order $$\alpha \in (0, N)$$ and $$\frac{N+\alpha }{N-2}$$ refers to the upper critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. By introducing the symmetric groups of Coxeter, we give a unified framework to construct saddle solutions with prescribed symmetric nodal structures. To overcome the difficulties arising from the lack of compactness, we give a fine analyzes on the profile decompositions of the symmetric Palais–Smale sequence. These results further feature the nonlocal nature of the Choquard equation, in contrast, the counterpart Yamabe equation $$-\Delta u=|u|^{\frac{4}{N-2}}u$$ can not permit such type of nodal solutions.

A new diffuse-interface approximation of the Willmore flow

Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that includes a rather simple but efficient penalization of the deviation from a quasi-one dimensional structure of the phase fields. We justify the approximation property by a Gamma convergence result for the energies and a matched asymptotic expansion for the flow. Ground states of the energy are shown to be one-dimensional, in contrast to the presence of saddle solutions for the usual diffuse approximation. Finally we present numerical simulations that illustrate the approximation property and apply our new approach to problems where the usual approach leads to an undesired behavior.

Euclidean black saddles and AdS4 black holes

We find new asymptotically locally AdS4 Euclidean supersymmetric solutions of the STU model in four-dimensional gauged supergravity. These “black saddles” have an S1× {Sigma}_{mathfrak{g}} boundary at asymptotic infinity and cap off smoothly in the interior. The solutions can be uplifted to eleven dimensions and are holographically dual to the topologically twisted ABJM theory on S1× {Sigma}_{mathfrak{g}} . We show explicitly that the on-shell action of the black saddle solutions agrees exactly with the topologically twisted index of the ABJM theory in the planar limit for general values of the magnetic fluxes, flavor fugacities, and real masses. This agreement relies on a careful holographic renormalization analysis combined with a novel UV/IR holographic relation between supergravity parameters and field theory sources. The Euclidean black saddle solution space contains special points that can be Wick-rotated to regular Lorentzian supergravity backgrounds that correspond to the well-known supersymmetric dyonic AdS4 black holes in the STU model.

Saddle solutions for the Choquard equation II

We study nodal solutions of saddle type for the Choquard equation −Δu+u=(Iα∗|u|p)|u|p−2uinRNwhere N≥3 and 2≤p<N+αN−2. The function Iα refers to the Riesz potential with order α∈(0,N). In a noncompact setting, we construct saddle solutions whose nodal domains are of conical shapes demonstrating polyhedral symmetry configurations in RN. More precisely, when N≥4, for given integers ℓ1,ℓ2≥2, this equation admits saddle solutions with its 4ℓ1ℓ2 nodal domains meeting at the origin. These among other saddle solutions not only reveal the nonlocal feature of the Choquard equation but also further complete the variational framework of constructing sign-changing solutions for the Choquard equation initiated in Ghimenti and Van Schaftingen (2016) and further developed in Xia and Wang (2019).

Saddle solutions for the Choquard equation

We study the Choquard equation $$\begin{aligned} -\Delta u + u= (I_\alpha *|u|^p )|u|^{p-2}u \quad \text { in }\;{\mathbb {R}}^N \end{aligned}$$ where $$N\ge 2$$ , $$I_\alpha $$ is the Riesz potential of order $$\alpha \in (0, N)$$ and $$2\le p<\frac{N+\alpha }{N-2}$$ . For any given integer $$\ell \ge 2$$ , we construct a saddle type nodal solution with its $$2\ell $$ nodal domains meeting at the origin. Moreover, this equation admits nodal solutions of higher dimensional nodal structure with exactly $$2^k$$ nodal domains for each $$1\le k\le N$$ . This is a new phenomenon for the Choquard equation which is nonlocal in nature when compared with its local counterpart the nonlinear Schrodinger equation.

A study of pattern forming systems with a fully nonlocal interaction kernel of possibly changing sign

A nonlocal model with a possibly sign changing kernel is proposed to study pattern formation problems in physical and biological systems with self-organization properties. One defines pre-solutions of a related linear problem and gives sufficient conditions for pre-solutions to be jump discontinuous solutions of the nonlocal model. Among a multitude of solutions a selection criterion is used to single out more significant critical solutions. These solutions are further classified into minimizing solutions, maximizing solutions and saddle solutions. Minimizing solutions are the most relevant for patterned states and they exist if the kernel changes sign. A non-negative kernel on the other hand behaves in some ways like the gradient term in the standard local model.

Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations

We study systems of elliptic equations \begin{document}$ -\Delta u(x)+F_{u}(x, u) = 0 $\end{document} with potentials \begin{document}$ F\in C^{2}({\mathbb{R}}^{n}, {\mathbb{R}}^{m}) $\end{document} which are periodic and even in all their variables. We show that if \begin{document}$ F(x, u) $\end{document} has flip symmetry with respect to two of the components of \begin{document}$ x $\end{document} and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on \begin{document}$ {\mathbb{R}}^{n} $\end{document} .

Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence

We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on ${\mathbb R}^3\times S^1$, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.

Existence of saddle solutions of a nonlinear elliptic equation involving p-Laplacian in more even-dimensional spaces

We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p > 2, -Δpu = f(u) in R2m for all dimensions satisfying 2m ≥ p, by using sub-supersolution method. The existence of saddle solutions of the above problem was known only in dimensions 2m ≥ 2p.