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In this paper, the initial value problem for singularly perturbed nonlinear holomorphic differential equations with a small parameter is considered in a complex domain, and the asymptotic behavior of solutions with respect to the parameter is studied. Under suitable conditions, a formal power series solution is constructed, and then it is proved that this formal solution is summable in a suitable direction. This guarantees the existence of a holomorphic solution that admits the formal power series solution as an asymptotic expansion of Gevrey type. It is then proved that this solution also has a Gevrey type asymptotic expansion with respect to the parameter. [ABSTRACT FROM AUTHOR]

This paper deals with the period function of the reversible quadratic centers X ν = − y (1 − x) ∂ x + (x + D x 2 + F y 2) ∂ y , where ν = (D , F) ∈ R 2. Compactifying the vector field to S 2 , the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle Π ν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of X ν that tend to Π ν 0 in the Hausdorff sense as ν → ν 0. This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν = (D , F) ∈ R 2 outside the segments { − 1 } × [ 0 , 1 ] and { 0 } × [ 0 , 2 ]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν ∈ R 2. In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u ″ (t) + A u (t) + b (t) u ′ (t) = 0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b (t) ∼ (1 + t) − β (− 1 < β < 1). In the case b (t) ≡ 1 , it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v ′ (t) + A v (t) = F n (t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b (t) v ′ (t) + A v (t) = 0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n -th order profile u ˜ n is given via v ′ (t) + m n (t) A v (t) = F n (t) which coefficient m n (time-scale) differs each other. [ABSTRACT FROM AUTHOR]

In this paper, we study heteroclinic bifurcation of limit cycles in a planar cubic near-Hamiltonian system by higher-order Melnikov functions. We compute the asymptotic expansion of the third-order Melnikov function near the heteroclinic loop L s and prove that this system can have five limit cycles around L s with proper perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via the Agrachev–Sarychev method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the defocusing Schrödinger (NLS) equation with a nonzero background i q t + q x x − 2 (| q | 2 − 1) q = 0 , q (x , 0) = q 0 (x) , lim x → ± ∞ ⁡ q 0 (x) = ± 1. Recently, for the space-time region | x / (2 t) | < 1 which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the N -soliton solutions for the NLS equation by using the ∂ ¯ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form (0.1) q (x , t) = T (∞) − 2 q s o l , N (x , t) + O (t − 1) , whose leading term is N-soliton and the second term O (t − 1) is a residual error from a ∂ ‾ -equation. In this paper, we are interested in the large-time asymptotics in the space-time region | x / (2 t) | > 1 which is outside the soliton region, but there will be two stationary points appearing on the jump contour R. We found an asymptotic expansion that is different from (0.1) (0.2) q (x , t) = e − i α (∞) (1 + t − 1 / 2 h (x , t)) + O (t − 3 / 4) , whose leading term is a nonzero background, the second t − 1 / 2 order term is from the continuous spectrum and the third term O (t − 3 / 4) is a residual error from a ∂ ‾ -equation. The above two asymptotic results (0.1) and (0.2) imply that the region | x / (2 t) | < 1 considered by Cuccagna and Jenkins is a fast decaying soliton solution region, while the region | x / (2 t) | > 1 considered by us is a slow decaying nonzero background region. • We obtain large-time asymptotics in the solitonless region |x/(2t)| > 1 for defocusing NLS equation with nonzero background. • This result is a complete supplement to the result recently obtained by Cuccagna and Jenkins. • | x / (2 t) | < 1 considered by Cuccagna and Jenkins is a fast decaying soliton region, while ours | x / (2 t) | > 1 is a slow decaying nonzero background region. [ABSTRACT FROM AUTHOR]

Let L u = u ″ + 2 A u ′ + B u be an ordinary differential operator with A and B are polynomials of any degree. In this paper we show that L is globally regular , i.e. L u ∈ S implies u ∈ S , for all tempered distribution u ∈ S ′ , if and only if the complex roots ξ ± (x) = i A (x) ± B (x) − A ′ (x) − (A (x)) 2 of the Weyl symbol σ L (x , ξ) = − ξ 2 + 2 i A (x) ξ + B (x) − A ′ (x) satisfy the condition lim | x | → + ∞ ⁡ | x Im ξ ± (x) | = + ∞. [ABSTRACT FROM AUTHOR]

Our aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends analytically on a small parameter, we derive a full asymptotic expansion of its Floquet exponents. Based on this, we prove that only the constant order Floquet exponents of multiplicity higher than one will be perturbed linearly. The required multiplicity can be achieved via folding of the system through certain choices of the periodicity of the coefficient matrix. Secondly, we apply such an asymptotic theory for the analysis of Floquet metamaterials. We provide a characterization of asymptotic exceptional points for a pair of subwavelength resonators with time-dependent material parameters. We prove that asymptotic exceptional points are obtained if the frequency components of the perturbations fulfill a certain ratio, which is determined by the geometry of the dimer of subwavelength resonators. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our preceding work on hyperbolic relaxation systems with characteristic boundaries of type I. Here we focus on the characteristic boundaries of type II, where the boundary is characteristic for the equilibrium system and is non-characteristic for the relaxation system. For this kind of characteristic initial-boundary-value problems (IBVPs), we introduce a three-scale asymptotic expansion to analyze the boundary-layer behaviors of the general multi-dimensional linear relaxation systems. Moreover, we derive the reduced boundary condition under the Generalized Kreiss Condition by resorting to some subtle matrix transformations and the perturbation theory of linear operators. The reduced boundary condition is proved to satisfy the Uniform Kreiss Condition for characteristic IBVPs. Its validity is shown through an error estimate involving the Fourier-Laplace transformation and an energy method based on the structural stability condition. [ABSTRACT FROM AUTHOR]

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3 n − 1 limit cycles. [ABSTRACT FROM AUTHOR]

In this paper we obtain higher order asymptotic expansions of solutions to the Cauchy problem of the linear damped wave equation in R n u t t − Δ u + u t = 0 , u (0 , x) = u 0 (x) , u t (0 , x) = u 1 (x) , where n ∈ N and u 0 , u 1 ∈ L 1 (R n) ∩ L 2 (R n). Established hyperbolic effects seem to be new in the sense that the order of obtained expansions depends on the spatial dimension. [ABSTRACT FROM AUTHOR]