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In this paper, we investigate the inviscid limit ν → 0 for time-quasi-periodic solutions of the incompressible Navier–Stokes equations on the two-dimensional torus T 2 , with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier–Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order O (ν 2) and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier–Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems. [ABSTRACT FROM AUTHOR]

In this paper, an infinite dimensional KAM theorem with unbounded perturbations and double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with bounded perturbations by Bambusi, Berti, Magistrelli, and us. As applications, for derivative nonlinear Schrödinger equation with periodic boundary conditions, quasi-periodic solutions around constant solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we establish the reducibility of a class of linear coupled quantum harmonic oscillator systems under time quasi-periodic, non-Hamiltonian, reversible perturbations. This essentially means that for most values of the frequency vector, these systems can be reduced to autonomous reversible systems with constant coefficients with respect to time. Our proof relies on an application of Kolmogorov–Arnold–Moser (KAM) theory for infinite dimensional reversible systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and stability of quasi-periodic solutions for the Navier–Stokes equation on irrational tori. We prove the persistence of quasi-periodic invariant tori under the small quasi-periodic external force. Furthermore, we obtain the orbital and asymptotic stability in the appropriate Sobolev space. Using the norm estimate of the heat propagator e t Δ α , we get the solutions convergence exponentially to the invariant tori as t → ∞ , with the rate of convergence O (e - ρ t) for any ρ ∈ (0 , 1) . [ABSTRACT FROM AUTHOR]

We prove the existence of steady space quasi-periodic stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel R × [ - 1 , 1 ] . These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash–Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin’s cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium. [ABSTRACT FROM AUTHOR]

In the present paper, we establish an infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with double normal frequencies. Applying it, we prove the existence of quasi-periodic solutions for one dimensional coupled nonlinear quantum harmonic oscillators (QHO) with a natural reversible structure. To compensate the lack of smoothing effect of perturbation, we introduce a class of vector fields with polynomial decay which extends the works of Grébert and Thomann (Commun Math Phys 307(2):383–427, 2011) for Hamiltonian QHO. To deal with the reversible, coupled perturbations in the equations, we also introduce a new class of generating vector fields during the KAM iteration. Moreover, the quasi-periodic solutions we obtain may not be linearly stable. This is obviously different from the result in Grébert and Thomann (2011) for Hamiltonian QHO. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of quasi-periodic response solutions (i.e., solutions that are quasi-periodic with the same frequencies as forcing term) for a class of forced reversible wave equations with derivative nonlinearity. The forcing frequency ω ∈ R 2 would be Liouvillean which is weaker than the usual Diophantine and Brjuno conditions. The derivative nonlinearity in the equation also leads to some difficulty in measure estimate. To overcome it, we also use the Töplitz–Lipschitz property of vector field. The proof is based on an infinite dimensional Kolmogorov–Arnold–Moser theorem for reversible systems. [ABSTRACT FROM AUTHOR]

In this paper, we establish quantitative Green's function estimates for some higher-dimensional lattice quasi-periodic (QP) Schrödinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential-type non-resonance conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the finite volume version of ( (1 2 −) )-Hölder continuity of the integrated density of states (IDS) for such QP Schrödinger operators. This gives an affirmative answer to Bourgain's problem in Bourgain (2000). [ABSTRACT FROM AUTHOR]

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the 1D wave equation system \[ \begin{cases} u_{1tt} - u_{1xx} +\sigma u_1 +u_1u_2^2 = 0 \\ u_{2tt} - u_{2xx} +\mu u_2 +u_1^2 u_2 = 0 \end{cases} \] { u 1 tt − u 1 xx + σ u 1 + u 1 u 2 2 = 0 u 2 tt − u 2 xx + μ u 2 + u 1 2 u 2 = 0 under Dirichlet boundary conditions, where $ 0 \lt \sigma \in [\sigma _1,\sigma _2]\subset [0,1], $ 0 < σ ∈ [ σ 1 , σ 2 ] ⊂ [ 0 , 1 ] , $ 0 \lt \mu \in [\mu _1,\mu _2]\subset [0,1] $ 0 < μ ∈ [ μ 1 , μ 2 ] ⊂ [ 0 , 1 ] are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system. [ABSTRACT FROM AUTHOR]

In this paper one-dimensional generalized eighth-order Boussinesq equation u tt - ∂ x 2 u + β ∂ x 4 u - ∂ x 6 u + ∂ x 8 u + (u 3) xx = 0 , β = ± 1 with the boundary conditions u (0 , t) = u (π , t) = u xx (0 , t) = u xx (π , t) = u xxxx (0 , t) = u xxxx (π , t) = 0 is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. [ABSTRACT FROM AUTHOR]

By modifying a Kuksin's estimate, the coefficients of a one-dimensional quasi-periodic linear Schrödinger equation can be reduced to constants, and thus, the existence of the quasi-periodic solutions is obtained. Moreover, for the reduced system, the boundedness, the blowup and the specific form of the quasi-periodic solutions are analyzed in detail, via the depiction of the phase portrait with respect to the pseudo-time in R 3 and the growth of solutions from the spectrum theory of the associated lattice Schrödinger operator numerically. The result is based on infinite-dimensional KAM theory and bifurcation theory, which is original. The reduction techniques can also be used to the Ginzburg–Landau equation, which can be applied to traffic flow hydrodynamics. The solitary wavelike quasi-periodic solutions obtained can be further refined and applied to optical soliton communication. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of quasi-periodic breathers in granular chains of coupled Duffing oscillators with Hertzian interaction potential, which is of a finite smoothness. Using Jackson–Moser–Zehnder analytic approximation technique and KAM iterations, not only do we obtain the existence and linear stability of quasi-periodic breathers, but also we provide concrete estimates on the localization rate, etc., depending on the smoothness order and the number of oscillating frequencies. [ABSTRACT FROM AUTHOR]

In this paper, we prove an infinite dimensional KAM (Kolmogorov–Arnold–Moser) theorem, which can be used to the KdV equations u t + ∂ x x x u − ε ∂ x f (ω t , x , u) = 0 , where ω = ξ ω ¯ , ω ¯ = (1 , α) is Liouvillean forced frequency and f is real analytic. We obtain a C∞ smooth response solution under zero mean-value periodic boundary conditions. The proof is based on a modified infinite dimensional KAM theory. [ABSTRACT FROM AUTHOR]

We prove the existence of time quasi-periodic vortex patch solutions of the 2 d -Euler equations in R 2 , close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -as well as the degeneracy of a normal frequency- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrödinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrödinger systems on two dimensional torus. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the existence of response solutions of quasi-periodic type for a class of quasi-periodically forced, non-Hamiltonian but reversible nonlinear beam equations. We do not suppose the basic frequency ω ∈ R 2 of the forcing term is Diophantine or Brjuno, and it might be Liouvillean, which is weaker than the Diophantine or Brjuno frequency. The proof is based on an improved Kolmogorov–Arnold–Moser (KAM) theorem for infinite-dimensional reversible systems with non-reducible normal form. [ABSTRACT FROM AUTHOR]

We prove spectral and strong dynamical localization in a random displacements model on the lattice, with singular (including Bernoulli) probability distribution of the displacements and an infinite-range site potential which can be asymmetric and sign-indefinite. Our techniques also apply to the models in continuous configuration spaces. [ABSTRACT FROM AUTHOR]

We study random displacements models with a long-range particle-media interaction potential 픲 ⁢ (r , θ) = 픣 ⁢ (θ) ⁢ r - A in polar coordinates, with a smooth function 픣 which can be sign-indefinite. Spectral and dynamical localization, with an asymptotically exponential decay of eigenfunction correlators, is proved under the optimal condition A > d . [ABSTRACT FROM AUTHOR]

This work studies a two-dimensional beam equation with a quintic nonlinear term and quasi-periodic forcing u tt + Δ 2 u + ε ϕ (t) h (u) = 0 , x ∈ T 2 , t ∈ R with periodic boundary conditions, where ε is a small positive parameter; ϕ (t) is a real analytic quasi-periodic function in t with frequency vector η = (η 1 , η 2 … , η n ∗) ⊂ [ ϱ , 2 ϱ ] n ∗ for a given positive integer n ∗ and some constant ϱ > 0 ; and h is a real analytic function of the form h (u) = c 1 u + c 5 u 5 + ∑ i ^ ≥ 6 c i ^ u i ^ , c 1 , c 5 ≠ 0. Firstly, the linear part of Hamiltonian system corresponding to the equation is transformed to constant coefficients by a linear quasi-periodic change of variables. Then, a symplectic transformation is used to convert the Hamiltonian system into an angle-dependent block-diagonal normal form, which can be achieved by selecting the appropriate tangential sites. Finally, it is obtained that a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation by developing an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional Hamiltonian systems. [ABSTRACT FROM AUTHOR]

In this paper we study discrete quasi-periodic operators with certain long-range hopping and monotonic meromorphic potentials. The hopping amplitude decays with the interparticle distance | n | as e - r log t (1 + | n |) ( t > 1 , r > 0 , n ∈ Z v ). By employing the KAM method, we prove such operators have pure point spectrum with eigenfunctions φ i i ∈ Z v obeying φ i (n) ≤ 2 e - r 2 log t (1 + | n - i |) for all i ∈ Z v , n ∈ Z v . [ABSTRACT FROM AUTHOR]

We prove the reducibility of quantum harmonic oscillators in R d perturbed by a quasi-periodic in time potential V (x , ω t) with logarithmic decay . By a new estimate built for solving the homological equation we improve the reducibility result by Grébert-Paturel(Annales de la Faculté des sciences de Toulouse : Mathématiques. 28 , 2019). [ABSTRACT FROM AUTHOR]

We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic dynamics is described by a highly resonant Floquet–Schrödinger equation, for which we show the existence of a periodic solution if the forcing frequency is away from a discrete set of resonances. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing, also at arbitrarily low non-resonant frequencies. The behaviour at the resonances remains a challenging open problem. [ABSTRACT FROM AUTHOR]

In this paper, the fractional nonlinear Schrödinger equation system with periodic boundary conditions is considered. I establish an abstract infinite-dimensional KAM theorem and apply it to fractional nonlinear Schrödinger equation system with real Fourier multiplier. By establishing a block diagonal normal form, we prove the existence of a class of Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite-dimensional invariant tori of an associated infinite-dimensional dynamic system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved. [ABSTRACT FROM AUTHOR]

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n - 1 with n ≥ 3 and d ≥ 2 . If ε ≪ 1 is the size of the initial datum, we prove that the lifespan T ε of solutions is O (ε - A (n - 2) - ) where A ≡ A (d , n) = 1 + 3 d - 1 when n is even and A = 1 + 3 d - 1 + max (4 - d d - 1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain T ε = O (ε - 6 - ) , much better than O (ε - 1) , the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of Δ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step. [ABSTRACT FROM AUTHOR]

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus T d , with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in H s (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for t → + ∞ , with an exponential rate of convergence O (e - α t) for any arbitrary α ∈ (0 , 1) . [ABSTRACT FROM AUTHOR]

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on T ∞ . [ABSTRACT FROM AUTHOR]

In this paper, two dimensional modified Boussinesq equation u tt + Δ 2 u + Δ (u 3) = 0 , x ∈ T 2 , t ∈ R under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. [ABSTRACT FROM AUTHOR]

In the present paper, we prove an infinite dimensional reversible Kolmogorov-Arnold-Moser (KAM) theorem. As an application, we study the existence of KAM tori for a class of two dimensional (2D) non-Hamiltonian completely resonant beam equations with derivative nonlinearities. The Birkhoff normal form theory is also used since there are no external parameters in the equations. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a beam equation with an x-dependent nonlinear term. We construct an analytic and symplectic transformation which changes the Hamiltonian to its Birkhoff normal form. However, the infinitely many coefficients of the Hamiltonian generating this transformation have small denominators. We prove that these denominators do not vanish for all indices and the transformation is canonical. Applying the normal form to a KAM theorem, it is proved that the equation admits quasi-periodic solutions with prescribed frequencies for any fixed potential constant. [ABSTRACT FROM AUTHOR]

Consider a class of 1-d Hamiltonian derivative nonlinear Schrödinger equations i ψ t = ∂ xx ψ + V ∗ ψ + i ∂ x ( ∂ ψ ¯ F (x , ψ , ψ ¯) ) , x ∈ T , where V ∈ Θ m Θ m is defined in (1.5)]. The nonlinearity of these equations includes (ψ x , ψ ¯ x) and depends on space variable x periodically, which means that the nonlinearity is unbounded (see Definition 1.1) and the momentum set (see Definition 2.2) of the corresponding Hamiltonian function is unbounded. In this paper, we obtain that for any potential V outside a small measure subset of Θ m , if the initial value is smaller than R ≪ 1 in p-Sobolev norm, then the corresponding solution to this equation is also smaller than 2R during a time interval (- c R - r ∗ , c R - r ∗ ) (for any given positive r ∗ ). The main methods are constructing Birkhoff normal forms to unbounded Hamiltonian systems which have unbounded momentum sets and using the special symmetry of the Hamiltonian functions to control p-Sobolev norms of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, the author establishes a reduction theorem for linear Schrödinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM (Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schrödinger operator possesses the property of pure point spectra and zero Lyapunov exponent. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the class of almost-periodically forced higher-dimensional beam equations u t t + (− Δ + μ) 2 u + ψ (ω t) u = 0 , μ > 0 , t ∈ R , x ∈ R d , subject to periodic boundary conditions, where ψ (ω t) is real analytic and almost-periodic in t. We show the existence of almost-periodic solutions for this equation under some suitable hypotheses. In the proof, we improve the KAM iteration to deal with the infinite-dimensional frequency ω = (ω 1 , ω 2 , ...) . [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of response solution (i.e., quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation: y tt (t , x) = μ y xxxx + y xx + y 3 + ε f (ω t , x) xx , x ∈ [ 0 , π ] , μ > 0 , subject to the hinged boundary conditions y (t , 0) = y (t , π) = y xx (t , 0) = y xx (t , π) = 0 , where ω = (1 , α) with α being any irrational numbers. The proof is based on a modified Kolmogorov–Arnold–Moser (KAM) iterative scheme. We will, at every step of KAM iteration, construct a symplectic transformation in a such way that the composition of these transformations reduce the original system to a new system which possesses zero as equilibrium. Note that we allow α to be any irrational numbers, and thus the frequency ω = (1 , α) is beyond Diophantine or Brjuno frequency, which we call as Liouvillean frequency. Moreover, the model under consideration is ill-posed and has complicated Hamiltonian structure. This makes homological equations appearing in KAM iteration are different from the ones in the classical infinite-dimensional KAM theory. The result obtained in this paper strengthens the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new class of derivative nonlinear Schrödinger equations with reversible nonlinearities of the form i u t + u xx + | u x | 4 u = 0 , (t , x) ∈ R × T. We obtain real analytic, linearly stable periodic solutions and quasi-periodic ones with two basic frequencies via infinite dimensional Kolmogorov–Arnold–Moser (KAM) theory for reversible systems. By investigating the gauge invariance and the compact form of vector fields, in our KAM iterative procedure, we remove the usual Diophantine restrictions on tangential frequencies and only use the Melnikov non-resonance conditions. In the proof, we also use Birkhoff normal form techniques due to the lack of external parameters in the equation above. [ABSTRACT FROM AUTHOR]

We consider the nonlinear wave equation (NLW) on the $d$-dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

This paper is concerned with one-dimensional derivative quintic nonlinear Schrödinger equation, i u t − u x x + i (| u | 4 u) x = 0 , x ∈ T. The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established. The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem. We mention that in the present paper the mean value of u does not need to be zero, but small enough, which is different from the assumption (1.7) in Geng-Wu [J. Math. Phys., 53, 102702 (2012)]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear Heisenberg Ferromagnetic chain equation under Dirichlet boundary conditions. By Taylor formula, the nonlinear Heisenberg Ferromagnetic chain equation can be described by the nonlinear Schrödinger type equation. Using an infinite dimensional KAM theorem for reversible system, we prove the existence of many n-dimensional invariant tori under sufficiently small perturbation and thus many time quasi-periodic solutions for the above equation. [ABSTRACT FROM AUTHOR]