New Lower Bounds of Spatial Analyticity Radius for the Kawahara Equation

On persistence of spatial analyticity for the dispersion-generalized periodic KdV equation

Potential symmetry generators and associated conservation laws of perturbed nonlinear equations

Partial differential equations (PDEs) involving perturbation terms with a small parameter often have less analytical structure, in particular, fewer symmetries and conservation laws, compared to the unperturbed PDEs. For such perturbed PDEs, approximate conservation laws can be consistently defined. The set of approximate conservation laws comprises equivalence classes where members of each class differ by a trivial approximate conservation law. Similar to exact ones, approximate conservation laws can be systematically constructed using the characteristic approach with approximate multipliers. Examples of new approximate conservation laws are presented for perturbed nonlinear heat and wave equations. For approximately variational problems, an analogue of the first Noether’s theorem relates approximate multipliers to evolutionary components of approximate local Lie symmetry generators. The multiplier method used to obtain approximate conservation laws includes the Noether approach and generalizes it to a non-variational system. The procedure to use approximate local symmetries to obtain new approximate conservation laws from known ones, in terms of fluxes and multipliers, is established and illustrated. It is shown that approximate conservation laws lead to potential systems that can be used to obtain new approximate potential symmetries of the given PDE system with a small parameter.

Persistence of spatial analyticity is studied for solutions of the generalized Korteweg‐de Vries (KdV) equation with higher order dispersionwhere , are integers. For a class of analytic initial data with a fixed radius of analyticity , we show that the uniform radius of spatial analyticity of solutions at time cannot decay faster than as . In particular, this improves a recent result due to Petronilho and Silva [Math. Nachr. 292 (2019), no. 9, 2032–2047] for the modified Kawahara equation (, ), where they obtained a decay rate of order . Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.

We examine the thermalisation/localization trade off in an interacting and disordered Kitaev model, specifically addressing whether signatures of many-body localization can coexist with the systems topological phase. Using methods applicable to finite size systems, (e.g. the generalized one-particle density matrix, eigenstate entanglement entropy, inverse zero modes coherence length) we identify a regime of parameter space in the vicinity of the non-interacting limit where topological superconductivity survives together with a significant violation of Eigenstate-Thermalisation-Hypothesis (ETH) at finite energy-densities. We further identify an anomalous behaviour of the von Neumann entanglement entropy which can be attributed to the prethermalisation-like effects that occur due to lack of hybridization between high-energy eigenstates reflecting an approximate particle conservation law. In this regime the system tends to thermalise to a generalised Gibbs ensemble (as opposed to the grand canonical ensemble). Moderate disorder tends to drive the system towards stronger hybridization and a standard thermal ensemble, where the approximate conservation law is violated. This behaviour is cutoff by strong disorder which obstructs many body effects thus violating ETH and reducing the entanglement entropy.

In terms of our new exact definition of partial Lagrangian and approximate Euler-Lagrange-type equation, we investigate the nonlinear wave equation with damping via approximate Noether-type symmetry operators associated with partial Lagrangians and construct its approximate conservation laws in general form.

The present work considers the Lie group analysis of a system of linear wave type perturbed systems. The methodology is based on finding approximate symmetry operators of a given system. Approximate conservation laws are found via an approximate version of Noether’s theorem. This is based on the modified Noether’s method provided by Ibragimov. Finally a numerical method is applied to solve the considered system.

The presence of (approximate) conservation laws can prohibit the fast relaxation of interacting many-particle quantum systems. We investigate this physics by studying the center-of-mass oscillations of two species of fermionic ultracold atoms in a harmonic trap. If their trap frequencies are equal, a dynamical symmetry (spectrum generating algebra), closely related to Kohn's theorem, prohibits the relaxation of center-of-mass oscillations. A small detuning $\delta\omega$ of the trap frequencies for the two species breaks the dynamical symmetry and ultimately leads to a damping of dipole oscillations driven by inter-species interactions. Using memory-matrix methods, we calculate the relaxation as a function of frequency difference, particle number, temperature, and strength of inter-species interactions. When interactions dominate, there is almost perfect drag between the two species and the dynamical symmetry is approximately restored. The drag can either arise from Hartree potentials or from friction. In the latter case (hydrodynamic limit), the center-of-mass oscillations decay with a tiny rate, $1/\tau \propto (\delta\omega)^2/\Gamma$, where $\Gamma$ is a single particle scattering rate.

Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems

In this paper, benefited some ideas of Wang [J. Geom. Anal. 33, 18 (2023)] and Dufera et al. [J. Math. Anal. Appl. 509, 126001 (2022)], we investigate persistence of spatial analyticity for solution of the higher order nonlinear dispersive equation with the initial data in modified Gevrey space. More precisely, using the contraction mapping principle, the bilinear estimate as well as approximate conservation law, we establish the persistence of the radius of spatial analyticity till some time δ. Then, given initial data that is analytic with fixed radius σ0, we obtain asymptotic lower bound σ(t)≥c|t|−12, for large time t ≥ δ. This result improves earlier ones in the literatures, such as Zhang et al. [Discrete Contin. Dyn. Syst. B 29, 937–970 (2024)], Huang–Wang [J. Differ. Equations 266, 5278–5317 (2019)], Liu–Wang [Nonlinear Differ. Equations Appl. 29, 57 (2022)], Wang [J. Geom. Anal. 33, 18 (2023)] and Selberg–Tesfahun [Ann. Henri Poincaré 18, 3553–3564 (2017)].

We consider the Cauchy problem for an equation of Korteweg–de Vries–Kawahara type with initial data in the analytic Gevrey spaces. By using linear, bilinear and trilinear estimates in analytic Bourgain spaces, we establish the local well-posedness of this problem. By using an approximate conservation law, we extend this to a global result in such a way that the radius of analyticity of solutions is uniformly bounded below by a fixed positive number for all time.

In this short chapter, we develop, via the method of matched asymptotic expansions, the small time asymptotic structure of the solution to IBVP (when n < m < 1) when the initial data, u 0(x), is a continuous, analytic, positive and monotone decreasing function in x≥ 0, with u o(x)→ 0 as x→ ∞. In particular, we consider the following cases: (a) Initial data that has an algebraic decay rate as x→ ∞ $${{u}_{0}}(x)\sim \left\{ \begin{gathered} {{u}_{\infty }}{{x}^{{ - \alpha }}} + EST(x)asx \to \infty , \hfill \\ {{{\tilde{u}}}_{0}} + \sum\nolimits_{{l = 1}}^{\infty } {{{{\tilde{u}}}_{l}}{{x}^{l}}asx \to 0 \to {{0}^{ + }},} \hfill \\ \end{gathered} \right.$$ (9.1) (b) Initial data that has an exponential decay rate as x→ ∞ $${{u}_{0}}(x)\sim \left\{ \begin{gathered} {{u}_{\infty }}{{x}^{{ - \beta }}}{{e}^{{ - \sigma x}}} + O\left[ {{{e}^{{ - f(x)}}}} \right]asx \to \infty , \hfill \\ {{{\tilde{u}}}_{0}} + \sum\nolimits_{{l = 1}}^{\infty } {{{{\tilde{u}}}_{l}}{{x}^{l}}asx \to {{0}^{ + }},} \hfill \\ \end{gathered} \right.$$ (9.2) for some f (x ) > O(x) as x ≥ ∞, where u∞, ũo, σ > 0 and β, ũl are constants.

Here we shall discuss analyticity results for several important partial differential equations. This includes the analytic regularity of sub-Laplacians under the finite type condition; the analyticity of the solution in both variables to the Cauchy problem for the Camassa–Holm equation with analytic initial data by using the Ovsyannikov theorem, which is a Cauchy–Kowalevski type theorem for nonlocal equations; the Cauchy problem for BBM with analytic initial data; the Cauchy problem for KdV with analytic initial data examining the evolution of uniform radius of spatial analyticity; and finally the time regularity of KdV solutions, which is Gevrey 3.

Recently it was shown by Z. Guo and the author independently that the Cauchy problem for the nonperiodic Korteweg-de Vries equation is locally well-posed in the Sobolev space of the critical regularity. Their proofs were based on the iteration argument in the Besov endpoint of the Bourgain space with some additional modification in low frequency, and it was conjectured that the Bourgain space with the Besov-type modification only does not restore the bilinear estimate which is essential to the iteration. In the present article we give the answer to this conjecture by constructing an exact counterexample to the bilinear estimate.

This paper concerns about the initial boundary-value problem (IBVP) with low regularities via the boundary integral operator method introduced by Bona–Sun–Zhang [Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations. J Math Pures Appl. 2018;109:1–66] on Bourgain-type space, i.e. X s , b − space. Unlike initial value problems, the choice of index b for IBVPs on X s , b − space will need to be less or equal to 1 2 . In addition, for different indexes (i.e. -\\frac {1}{2} $ ]]> s > − 1 2 and s ≤ − 1 2 ), variety of extensions on the boundary integral operator in X s , b − space will be adapted which provides different requirements on b accordingly (i.e. b ≤ 1 2 and b < 1 2 ). In this article, we use the nonlinear Schrödinger equation with different quadratic nonlinearities as examples, 0, \\\\ u(x, 0)=\\varphi(x),\\quad u(0, t)=h(t), \\end{array}\\right. \\] ]]> { i u t + u xx + N i ( u , u ¯ ) = 0 , x , t > 0 , u ( x , 0 ) = φ ( x ) , u ( 0 , t ) = h ( t ) , where N 1 ( u , u ¯ ) = u u ¯ , N 2 ( u , u ¯ ) = u ¯ 2 , to establish well-posed theories for -\\frac {1}{4} $ ]]> s > − 1 4 and -\\frac {3}{4} $ ]]> s > − 3 4 respectively, based on bilinear estimates with different values of b in Bourgain space X s , 1 2 and X s , b ( b < 1 2 ) accordingly. Though similar results have been shown in [Cavalcante M. The initial boundary-value problem for some quadratic nonlinear Schrödinger equations on the half line, 2016] for b < 1 2 , based on Colliander–Kenig–Holmer's approach (see Holmer J. [The initial boundary-value problem for the 1-D nonlinear Schrödinger equation on the half-line. Differ Int Equ. 2005]; Colliander JE, Kenig CE. [The generalized Korteweg–de Vries equation on the half line. arXiv Mathematics e-prints, page math/0111294, Nov. 2001]), our idea follows from Tao's multiplier method [Tao T. Multi-linear weighted convolution of L 2 functions and applications to nonlinear dispersive equations, 2004] and the conclusion is slightly stronger.