Continuum Memory System (CMS)

• Continuum Memory System (CMS) is a framework that rigorously bridges discrete nonlinear Schrödinger models with long-range interactions and memory effects to their continuous dispersive PDE limits.
• It employs fractional Caputo derivatives and the fractional Laplacian to encode slowly decaying memory kernels and nonlocal-in-time dynamics, enabling precise convergence analysis.
• The system leverages Laplace and Fourier transforms to derive explicit propagator formulas and convergence rates critical for stability and well-posedness in functional analysis.

A Continuum Memory System (CMS) refers to the continuum limit of discrete nonlinear Schrödinger (NLS) models on lattices with long-range interactions and intrinsic memory effects governed by fractional time derivatives. This framework rigorously characterizes the convergence of such discrete models—ubiquitous in the microscopic study of charge and energy transport in biomolecular chains—to continuous nonlinear dispersive PDEs with memory terms, particularly via the Caputo fractional derivative. The defining feature of the CMS is the persistence of integral, slowly decaying memory kernels in the governing equations, leading to nonlocal-in-time dynamics inherited from the discrete system even as the lattice mesh size vanishes.

1. Mathematical Formulation of the Continuum Memory Equation

The CMS limit equation for $x\in\mathbb R$, $t>0$, with initial datum $u_0(x)\in H^s(\mathbb R)$ and parameters $\alpha\in(1,2)$, $\beta\in(0,1)$, $\sigma=\alpha/\beta$ reads: $$\begin{cases} i^{\beta}\,\partial_t^{\beta}u(x,t)=(-\Delta)^{\alpha/2}u(x,t)\pm|u(x,t)|^{p-1}u(x,t),\ u(x,0)=u_0(x). \end{cases}$$ Here, $\partial_t^{\beta}u$ is the Caputo time derivative, defined by

$$\partial_t^{\beta}u(t) = \frac{1}{\Gamma(1-\beta)} \int_{0}^{t} \frac{\partial_s u(s)}{(t-s)^{\beta}} ds,$$

which manifests an integral memory operator with weakly decaying kernel $K_C(t) = t^{-\beta}/\Gamma(1-\beta)$. The fractional Laplacian $(-\Delta)^{\alpha/2}$ acts via the Fourier multiplier: $$\widehat{(-\Delta)^{\alpha/2}f}(\xi)=|\xi|^{\alpha}\widehat f(\xi)$$. The nonlinearity adopts the standard power form, $F(u)=\pm|u|^{p-1}u$ for odd integer $p\geq3$.

The system can be recast in pure memory form isolating the Caputo derivative: $$i^\beta\,\frac{1}{\Gamma(1-\beta)}\int_0^t \frac{\partial_su(x,s)}{(t-s)^\beta} ds +(-\Delta)^{\alpha/2}u(x,t)\pm|u(x,t)|^{p-1}u(x,t) =0,$$ and its Duhamel formulation involves propagators built from Mittag–Leffler functions, for example: $$u(t) = L_t u_0 \pm i^{-\beta} \int_0^t (t-s)^{\beta-1}E_{\beta,\beta}\bigl(i^{-\beta}(t-s)^{\beta}(-\Delta)^{\alpha/2}\bigr)F(u(s))\,ds,$$ with $L_t$ determined by $E_\beta(z)$, the Mittag–Leffler function: $E_\beta(z)=\sum_{k\geq0}z^k/\Gamma(k\beta+1)$.

2. Derivation from Discrete Models

The CMS framework is based on the rigorous passage from discrete models defined on infinite lattices $h\mathbb Z$ (mesh size $h>0$) to their continuum counterpart as $h\to0$. The discrete wave-function $u_h(t,mh)$, $m\in\mathbb Z$, satisfies: $$i^\beta \partial_t^\beta u_h = (-\Delta_h)^{\alpha/2} u_h \pm \Pi_h R_h(|u_h|^{p-1}u_h), \quad u_h(0) = \Pi_h f_{2h},$$ where $(-\Delta_h)^{\alpha/2}$ is the discrete fractional Laplacian: $$(-\Delta_h)^{\alpha/2}u_h(mh) = h^{-\alpha}\sum_{n\ne m} \frac{u_h(mh)-u_h(nh)}{|m-n|^{1+\alpha}},$$ with Fourier symbol $h^{-\alpha}w(\xi)$, $w(\xi)=2\sum_{n\geq1}(1-\cos(n\xi))/n^{1+\alpha}$, and as $h\rightarrow0$, $h^{-\alpha}w(h\xi)\to|\xi|^{\alpha}$. Operators $\Pi_h$ and $R_h$ provide essential interpolation and filtering to control lattice resonances.

The derivation utilizes Laplace transforms in time and Fourier in space, yielding linear propagator expressions in terms of Mittag–Leffler functions. The scaling $$\phi_h(\xi)=h^{-\alpha/\beta}w(h\xi)^{1/\beta}\rightarrow|\xi|^{\alpha/\beta}$$ captures the transition of symbols. Uniform smoothing and maximal-function estimates are essential to control convergence. As $h\to0$, the interpolated solution $p_hu_h$ converges to the continuum solution $u$ strongly in relevant Sobolev norms.

3. Convergence Theorem and Rates

The central theoretical result is the precise identification of convergence rates between the discrete and continuum systems. With fixed $\alpha\in(1,2)$, $\beta\in[1/2,1)$, $\sigma=\alpha/\beta$, $p\ge3$ odd, and regularity indices $s=1/2-1/(2(p-1))$, $\widetilde s=\max\{s+\sigma-\alpha,1/2\}<1$, and assuming $u_0\in H^{\widetilde s+}(\mathbb R)$:

• The continuum CMS has a unique solution $u\in C([0,T]; H^s(\mathbb R))$ for some $T>0$.
• The discrete lattice system admits a unique solution $u_h$ on $h\mathbb Z$ for $h>0$.
• The piecewise-linear interpolant $p_hu_h$ satisfies

$$\|p_hu_h - u\|_{L^\infty_t H^s_x} \longrightarrow 0\quad \text{as}~h\to0,$$

with explicit convergence rate

$$\sup_{0\leq t\leq T} \|p_hu_h(t)-u(t)\|_{H^s} \lesssim h^{2-\alpha-},$$

where “$-$” indicates an arbitrarily small loss.

4. Functional Analysis Framework and Key Estimates

Analysis of the CMS employs Banach spaces adapted to the loss/gain of derivatives induced by the fractional time derivative. The relevant space is

$$X^s_T = \left\{ v\in C([0,T];H^s_x)~|~ \|\langle\nabla\rangle^{s+\sigma-\alpha}v\|_{L^\infty_xL^2_t} +\|\langle\nabla\rangle^s v\|_{L^\infty_tL^2_x} + \|v\|_{L^{2(p-1)}_xL^\infty_t}<\infty \right\},$$

encoding both smoothing gains and losses of regularity. The foundational estimates include:

• Smoothing Effects: For the propagator $L_t$, $$\|\langle\nabla\rangle^\delta L_t f\|_{L^\infty_x L^2_t}\lesssim \|f\|_{H^s_x}$$ for $\delta\in[s+\sigma-\alpha, \sigma/2 - 1/(2(p-1)))$, along with discrete analogs.
• Maximal Function Estimates: $$\|e^{-it|\nabla|^\sigma}f\|_{L^p_xL^\infty_t}\lesssim \||\nabla|^s f\|_{L^2_x}$$ for $s=1/2-1/p$, $p\geq4$, including discrete versions.
• Sobolev and interpolation inequalities: $\|u_h\|_{\ell^\infty_h}\lesssim\|u_h\|_{H^s_h}$ for $s>1/2$; $\|p_hf_h\|_{H^s_x} \lesssim \|f_h\|_{H^s_h}$.

These analytic tools enable control of solution regularity and robustness under the limit $h\to0$ necessary for rigorous convergence.

5. Dynamical Properties and Well-Posedness

The CMS continuum equation exhibits local well-posedness in $H^s(\mathbb R)$ for the prescribed range of parameters, with the Caputo derivative causing a “loss” of $\sigma-\alpha$ derivatives, counteracted precisely by smoothing effects of the fractional Laplacian. Existence, uniqueness, continuous dependence, and persistence of regularity are established via a fixed-point argument in the Banach space $X^s_T$. The analytic theory extends mutatis mutandis to generalizations: higher-dimensional fractional Schrödinger equations with memory, broader classes of power nonlinearities, and alternative dispersive operators of fractional type.

6. Memory Kernels and Long-Time Behavior

The core memory mechanism in CMS arises from the Caputo derivative’s kernel $K_C(t) = t^{-\beta}/\Gamma(1-\beta)$—a convolution operator yielding slowly decaying memory tails. In the full Duhamel representation, the nonlinear memory kernel is

$$K_N(t,\xi) = t^{\beta-1}E_{\beta,\beta}(i^{-\beta}t^\beta|\xi|^\alpha),$$

introducing oscillatory, slowly decaying tails in time, with asymptotics (for $$\arg z\in[-\tfrac{\beta\pi}{2},\tfrac{\beta\pi}{2}]$$): $$E_\beta(z)\sim \frac{1}{\beta}e^{z^{1/\beta}}, \quad E_{\beta,\beta}(z)\sim \frac{1}{\beta}z^{1/\beta-1}e^{z^{1/\beta}}, \quad (|z|\rightarrow\infty).$$ Such features encode long-range temporal dependencies in the evolution, persisting even in the continuum PDE limit and corresponding to the influence of discrete interaction kernels $J_{n-m}\sim|n-m|^{-1-\alpha}$. This memory term is intrinsic to the limiting equation and is not erased under transition to the continuum.

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The continuum memory system formalism synthesizes discrete fractional lattice dynamics and fractional-time dispersive PDE analysis, providing sharp convergence rates, an explicit analytic framework, and foundational estimates essential for well-posedness and stability. Its development leverages techniques of smoothing, maximal function estimates, and functional analysis, and opens avenues for generalization to broader classes of dispersive memory PDEs (Grande, 2019).