Stable blow-up dynamics in the -critical and -supercritical generalized Hartree equation

We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: First, we consider the -critical case in dimensions and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate , but also the log-log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow-up regime in the -critical nonlinear Schrödinger equation. In this setting, we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled , a ground state solution of the elliptic equation . We also consider the -supercritical case in dimensions . We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS -supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level , and thus, numerically observable (unlike the -critical case). In summary, we find that the results are similar to the behavior of stable self-similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in the stable formation of singularities.     

The regularity of the multiple higher-order poles solitons of the NLS equation

Based on the inverse scattering method, the formulae of one higher-order pole solitons and multiple higher-order poles solitons of the nonlinear Schrödinger equation (NLS) equation are obtained. Their denominators are expressed as , where is a matrix frequently constructed for solving the Riemann-Hilbert problem, and the asterisk denotes complex conjugate. We take two methods for proving is invertible. The first one shows matrix is equivalent to a self-adjoint Hankel matrix , proving . The second one considers the block-matrix form of , proving . In addition, we prove that the dimension of is equivalent to the sum of the orders of pole points of the transmission coefficient and its diagonal entries compose a set of basis.     

Stability of equilibrium solutions of a double power reaction-diffusion equation with a Dirac interaction

In this paper, information about the instability of equilibrium solutions of a nonlinear family of localized reaction-diffusion equations in dimension one is provided. More precisely, explicit formulas to the equilibrium solutions are computed and, via analytic perturbation theory, the exact number of positive eigenvalues of the linear operator associated to the stability problem is analyzed. In addition, sufficient conditions for blow up of the solutions of the equation are also discussed.     

Sommerfeld-type integrals for discrete diffraction problems

Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).     

变角度纤维复合材料层合斜板的颤振分析

假设纤维方向角沿层合板的长度方向线性变化，研究了变角度纤维复合材料层合斜板的颤振．通过坐标变换将斜板变换为正方形板，采用层合板表面连续变化的速度环量来模拟空气对其的作用，速度环量分布利用Cauchy积分公式计算．建立了系统的Lagrange方程并采用Ritz法得到了层合板的自振频率和颤振/不稳定性分离临界速度．通过数值算例验证了本文模型和方法的正确性和收敛性，分析了各个铺层内纤维方向角的变化对自振频率和颤振/不稳定性分离临界速度的影响．研究结果表明，通过纤维的变角度铺设，可有效地提高层合板的基频和颤振/不稳定性分离临界速度．经合理设计的变角度复合材料层合板具有抑制颤振的作用．     

Geometrical methods for analyzing the optimal management of tipping point dynamics

Natural resources are not infinitely resilient and should not be modeled as being such. Finitely resilient resources feature tipping points and history dependence. This paper provides a didactical discussion of mathematical methods that are needed to understand the optimal management of such resources: viscosity solutions of Hamilton–Jacobi–Bellman equations, the costate equation and the associated canonical equations, exact root counting, and geometrical methods to analyze the geometry of the invariant manifolds of the canonical equations. Recommendations for Resource Managers

• Management of natural resources has to take into account the possible breakdown of resilience and induced regime shifts.
• Depending on the characteristics of the resource and on its present and future economic importance, either for all initial states the same kind of management policy is optimal, or the type of the optimal management policy depends on the initial state.
• Modeling should reflect the finiteness of the data.

     

Multiple bright and dark soliton solutions in three component spinor Bose-Einstein condensates

We investigate a quasi one dimensional spin-1 Bose-Einstein Condensates (BEC) in the absence of an external confinement governed by a system of three coupled Gross-Pitaevskii (GP) equation. Based on the Lax-pair, we construct one soliton solution employing gauge transformation method. In addition, the multiple bright and dark soliton solutions are obtained by properly choosing amplitude dependent parameter in the Lax-pair. The results of the paper emphasizes the richness in the structure of soliton solutions admitted by the spin components, a phenomenon which has never been brought out to the fore. We have also extended the gauge transformation method to generate two soliton solutions.     

Generalization of the Kelvin equation for arbitrarily curved surfaces

Capillary condensation, which takes place in confined geometries, is the first-order vapor-to-liquid phase transition and is explained by the Kelvin equation, but the equation's applicability for arbitrarily curved surface has been long debated and is severe problem. Recently, we have proposed generic dynamic equations for moving surfaces. Application of the equations to the vapor/fluid interfaces in chemical equilibrium conditions nearly trivially solves the generalization problem for the Kelvin equation. The equations are universally true for any surfaces: atomic, molecular, micro or macro scale, real or virtual, Riemannian or pseudo-Riemannian, active or passive.