On the controllability for the Khokhlov–Zabolotskaya–Kuznetsov-like equation

Recalling the proprieties of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, we prove the controllability of moments result for the linear part of the KZK equation and its non-linear perturbation.     

On the constraints problem for the Einstein–Yang–Mills–Higgs system

We provide proofs of some key propositions that were used in previous work by Dossa and Tadmon dealing with the characteristic initial value problem for the Einstein–Yang–Mills–Higgs (EYMH) system. The aforesaid proofs were missing, making the considered work difficult to understand. This work is presented with a view to have an almost self-contained paper. With this respect we completely recall the process of constructing initial data for the EYMH system on two intersecting smooth null hypersurfaces as done in the work of Dossa and Tadmon mentioned above. This is achieved by successfully adapting the hierarchical method set up by Rendall to solve the same problem for the Einstein equations in vacuum and with perfect fluid source. Many delicate calculations and expressions are given in details so as to address, in a forthcoming work, the issue of global resolution of the characteristic initial value problem for the EYMH system. The method obviously applies to the Einstein–Maxwell and the Einstein-scalar field models as well.     

On the spectrum of the Page and the Chen–LeBrun–Weber metrics

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen–LeBrun–Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the estimation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue.     

Proof of the Andrews–Dyson–Rhoades conjecture on the spt-crank

The spt-crank of a vector partition, or an S  -partition, was introduced by Andrews, Garvan and Liang. Let _NS(m,n)$N_S(m,n)$ denote the net number of S-partitions of n with spt-crank m, that is, the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is odd minus the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is even. Andrews, Dyson and Rhoades conjectured that _{NS(m,n)}m${\{N_S(m,n)\}}_m$ is unimodal for any n  , and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and crank of ordinary partitions, which implies _NS(m,n)≥_NS(m+1,n)$N_S(m,n)\ge N_S(m+1,n)$ for sufficiently large n and fixed m. In this paper, we introduce a representation of an ordinary partition, called the m-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.     

Stability of the nonconstant stationary solution to the Poisson–Nernst–Planck–Navier–Stokes equations

We study the three-dimensional Cauchy problem of the Poisson–Nernst–Planck–Navier–Stokes equations. We first show that the corresponding stationary system has a unique semi-trivial solution under a general doping profile. Under initial small perturbations around such the semi-trivial stationary solution and small doping profile, we obtain the unique global-in-time solution to the non-stationary system. Moreover, we prove the asymptotic convergence of the solution toward the semi-trivial stationary solution as time tends to infinity.     

On the Hald–Gesztesy–Simon Theorem

The method (Martynyuk and Pivovarchik, Inverse Probl. 26(3):035011, 2010) of recovering the potential of the Sturm–Liouville equation on a half of the interval by the spectrum of a boundary value problem and by the restriction of the potential onto the other half of the interval is used for treating the missing eigenvalue problem (Trans. Am. Math. Soc. 352:2765–3789, 2000, J. R. Astr. Soc. 62:41–48, 1980, J. Math. Pures Appl. 91:468–475, 2009, J. Math. Soc. Japan 38:39–65, 1986). The latter arises in the case of the half-inverse (Hochstadt–Lieberman) problem with Robin boundary conditions and lies in the fact that in many cases all the eigenvalues but one are needed to recover the potential and the Robin condition at one of the ends.     

On the Lazarev–Lieb extension of the Hobby–Rice theorem

O. Lazarev and E.H. Lieb proved that, given _f1,…,_fn∈L¹([0,1];C)$f_1,\dots ,f_n\in L^1([0,1];\mathbb{C})$, there exists a smooth function Φ$\Phi$ that takes values on the unit circle and annihilates span{_f1,…,_fn}$\text{span}\{f_1,\dots ,f_n\}$. We give an alternative proof of that fact that also shows the W^1,1$W^{1,1}$ norm of Φ$\Phi$ can be bounded by 5πn+1$5\pi n+1$. Answering a question raised by Lazarev and Lieb, we show that if p>1$p>1$ then there is no bound for the W^1,p$W^{1,p}$ norm of any such multiplier in terms of the norms of _f1,…,_fn$f_1,\dots ,f_n$.     

On the Aleksandrov–Bakelman–Pucci estimate for the infinity Laplacian

We prove L ^∞ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and p-Laplacian, namely $$\begin{array}{ll} -\Delta_p^N u=f\quad{\rm for } \; n < p \leq\infty.\end{array}$$ We are able to provide a stable family of results depending continuously on the parameter p. We also prove the failure of the classical Alexandrov–Bakelman–Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.     

Heat Flow for the Yang–Mills–Higgs Field and the Hermitian Yang–Mills–Higgs Metric

For a parameter > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the -stability of (E, ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.     

Riemann–Hilbert approach for the Camassa–Holm equation on the line

We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation $u_t?u_{txx}+2\omega u_x+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}$ on the line (CH). We show that: (i) for all $\omega >0$, the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann–Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for $\omega =0$. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On the Carey–Helton–Howe–Pincus trace formula

In this article, we give a new proof of the Carey–Helton–Howe–Pincus trace formula using Kato's theory of “relatively-smooth” operators and Krein's trace formula.     

On the Hamilton–Poisson realizations of the integrable deformations of the Maxwell–Bloch equations

In this note, we construct integrable deformations of the three-dimensional real valued Maxwell–Bloch equations by modifying their constants of motions. We obtain two Hamilton–Poisson realizations of the new system. Moreover, we prove that the obtained system has infinitely many Hamilton–Poisson realizations. Particularly, we present a Hamilton–Poisson approach of the system obtained considering two concrete deformation functions.     

Inversion of the Cauchy–Bunyakovskii–Schwarz inequality

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.     

Multiple Vortices in the Aharony–Bergman–Jafferis–Maldacena Model

Vortices in non-Abelian gauge field theory play important roles in confinement mechanism and are governed by systems of nonlinear elliptic equations of complicated structures. In this paper, we present a series of existence and uniqueness theorems for multiple vortex solutions of the BPS vortex equations, arising in the dual-layered Chern–Simons field theory developed by Aharony, Bergman, Jafferis, and Maldacena, over ${\mathbb{R}^2}$ and on a doubly periodic domain. In the full-plane setting, we show that the solution realizing a prescribed distribution of vortices exists and is unique. In the compact setting, we show that a solution realizing n prescribed vortices exists over a doubly periodic domain ${\Omega}$ if and only if the condition $$n < \frac{\lambda |\Omega|}{2 \pi}$$ holds, where ${\lambda >0 }$ is the Higgs coupling constant. In this case, if a solution exists, it must be unique. Our methods are based on calculus of variations.