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1. A rearrangement minimization problem corresponding to p -Laplacian equation

   https://doi.org/10.1051/cocv/2022004  

   Kao, Chiu-Yen; Mohammadi, Seyyed Abbas (January 2022, ESAIM: Control, Optimisation and Calculus of Variations)

   In this paper a rearrangement minimization problem corresponding to solutions of the p -Laplacian equation is considered. The solution of the minimization problem determines the optimal way of exerting external forces on a membrane in order to have a minimum displacement. Geometrical and topological properties of the optimizer is derived and the analytical solution of the problem is obtained for circular and annular membranes. Then, we find nearly optimal solutions which are shown to be good approximations to the minimizer for specific ranges of the parameter values in the optimization problem. A robust and efficient numerical algorithm is developed based upon rearrangement techniques to derive the solution of the minimization problem for domains with different geometries in ℝ 2 and ℝ 3 . 

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2. Notes on solutions of KZ equations modulo $p^s$ and $$p$$-adic limit $$s\to\infty$$

   Alexander Varchenko (January 2022, Contemporary mathematics)

   We consider the KZ equations over C in the case, when the hypergeometric solutions are hyperelliptic integrals of genus g. Then the space of solutions is a 2g-dimensional complex vector space. We also consider the same equations modulo ps, where p is an odd prime and s is a positive integer, and over the field Q_p of p-adic numbers. We construct polynomial solutions of the KZ equations modulo ps and study the space Mps of all constructed solutions. We show that the p-adic limit of Mps as s→∞ gives us a g-dimensional vector space of solutions of the KZ equations over Qp. The solutions over Qp are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the case g=1 of elliptic integrals. The p-adic limit of Mps as s→∞ gives us a one-dimensional space of solutions over Qp at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over Qp defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Qp. We describe the Frobenius transformations of solutions of the KZ equations for g=1 and then recover the unit roots of the zeta functions of the elliptic curves defined by the equations y2=βx(x−1)(x−α) over the finite field Fp. Here α,β∈F×p,α≠1 

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3. Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $$({\rm D}_7)$$ Equation

   https://doi.org/10.3842/SIGMA.2024.008  

   Buckingham, Robert J; Miller, Peter D (January 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$$_7$$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation. 

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4. Lump solutions to a (2+1)-dimensional fifth-order KdV-like equation

   Batwa, Sumayah; Ma, Wen-Xiu (July 2018, Advances in Mathematical Physics (Print))

   A (2+1)-dimensional fifth-order KdV-like equation is introduced through a generalized bilinear equation with the prime number . The new equation possesses the same bilinear form as the standard (2+1)-dimensional fifth-order KdV equation. By Maple symbolic computation, classes of lump solutions are constructed from a search for quadratic function solutions to the corresponding generalized bilinear equation. We get a set of free parameters in the resulting lump solutions, of which we can get a nonzero determinant condition ensuring analyticity and rational localization of the solutions. Particular classes of lump solutions with special choices of the free parameters are generated and plotted as illustrative examples. 

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5. Master curves for FENE-P fluids in steady shear flow

   https://doi.org/10.1016/j.jnnfm.2022.104944  

   Yamani, Sami; McKinley, Gareth H. (March 2023, Journal of Non-Newtonian Fluid Mechanics)

   Poole, Robert J (Ed.)

   The FENE-P (Finitely-Extensible Nonlinear Elastic) dumbbell constitutive equation is widely used in simulations and stability analyses of free and wall-bounded viscoelastic shear flows due to its relative simplicity and accuracy in predicting macroscopic properties of dilute polymer solutions. The model contains three independent material parameters, which expressed in dimensionless form correspond to a Weissenberg number (Wi), i.e., the ratio of the dumbbell relaxation time scale to a characteristic flow time scale, a finite extensibility parameter (L), corresponding to the ratio of the fully extended dumbbell length to the root mean square end-to-end separation of the polymer chain under equilibrium conditions, and a solvent viscosity ratio. An exact solution for the rheological predictions of the FENE-P model in steady simple shear flow is available [Sureshkumar et al., Phys Fluids (1997)], but the resulting nonlinear and nested set of equations do not readily reveal the key shear-thinning physics that dominates at high Wi as a result of the finite extensibility of the polymer chain. In this note we review a simple way of evaluating the steady material functions characterizing the nonlinear evolution of the polymeric contributions to the shear stress and first normal stress difference as the shear rate increases, provide asymptotic expansions as a function of Wi , and show that it is in fact possible to construct universal master curves for these two material functions as well as the corresponding stress ratio. Steady shear flow experiments on three highly elastic dilute polymer solutions of different finite extensibilities also follow the identified master curves. The governing dimensionless parameter for these master curves is Wi/L and it is only in strong shear flows exceeding Wi/L > 1 that the effects of finite extensibility of the polymer chains dominate the evolution of polymeric stresses in the flow field. We suggest that reporting the magnitude of Wi/L when performing stability analyses or simulating shear-dominated flows with the FENE-P model will help clarify finite extensibility effects. 

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