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Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$ Lax pair, in which the coefficient matrices are polynomials of spectral parameter. The first-order monic Darboux matrix is constructed explicitly and its classification theorem is presented. Then by using the solutions of the corresponding adjoint Lax pair, the $n$-order monic Darboux matrix and its inverse, both sharing the same unique pole, are derived explicitly. Further, a theorem is proposed to describe the invariance of Darboux matrix regarding pole distributions in Darboux matrix and its inverse. Finally, a unified theorem is offered to construct formal Darboux transformation in general form. All Darboux matrices expressible as the product of $n$ first-order monic Darboux matrices can be constructed in this way. The nonlocal focusing NLS equation, the focusing NLS equation and the Kaup-Boussinesq equation are taken as examples to illustrate the application of these Darboux transformations.

For the KdV equation with well-type initial value, the interaction between the trial soliton and the mean field is studied. The well initial value will lead to the appearance of rarefaction wave and dispersion shock wave, and there will be a linear wave region after a long time. The interaction between trial soliton and mean field is described within the framework of Whitham modulation theory, and the trajectory of soliton is given. The predicted soliton amplitude and phase changes are numerically confirmed, verifying the correctness of the theoretical analysis.

This paper employs the Riemann-Hilbert problem to provide a comprehensive analysis of the asymptotic behavior of the high-genus Korteweg-de Vries soliton gases. It is demonstrated that the two-genus soliton gas is related to the two-phase Riemann-Theta function as \(x \to +\infty\), and approaches to zero as \(x \to -\infty\). Additionally, the long-time asymptotic behavior of this two-genus soliton gas can be categorized into five distinct regions in the \(x\)-\(t\) plane, which from left to right are rapidly decay, modulated one-phase wave, unmodulated one-phase wave, modulated two-phase wave, and unmodulated two-phase wave. Moreover, an innovative method is introduced to solve the model problem associated with the high-genus Riemann surface, leading to the determination of the leading terms, which is also related with the multi-phase Riemann-Theta function. A general discussion on the case of arbitrary \(N\)-genus soliton gas is also presented.

For the KdV equation with box type initial data, the interaction between a trial soliton and large-scale dispersive mean flow is studied theoretically and numerically. The pure box initial value can cause rarefaction wave and dispersive shock wave, and can create an area of soliton train. The key to the interaction of soliton and mean flow is that the dynamic evolutions of the mean flow and the local soliton can be described by the same modulation system. The soliton modulation system is derived from the degenerations of the two-genus Whitham modulation system. Considering the influence of rarefaction wave, dispersive shock wave and soliton train on the trial soliton, in the framework of Whitham modulation theory, the equation describing the soliton trajectory and the changes in amplitude and phase shift are given explicitly. The predicted results are compared with the numerical simulations, which verifies the corrections of the theoretical analysis. The exotic interaction phenomena between soliton and mean flow found in this work have broad applications to shallow water soliton propagations and real soliton experiments in fluid dynamics.

In this paper, the renowned Riemann-Hilbert method is employed to investigate the initial value problem of Tzitz\'eica equation on the line. Initially, our analysis focuses on elucidating the properties of two reflection coefficients, which are determined by the initial values. Subsequently, leveraging these reflection coefficients, we construct a Riemann-Hilbert problem that is a powerful tool to articulate the solution of the Tzitz\'eica equation. Finally, the nonlinear steepest descent method is applied to the oscillatory Riemann-Hilbert problem, which enables us to delineate the long-time asymptotic behaviors of solutions to the Tzitz\'eica equation across various regions. Moreover, it is shown that the leading-order terms of asymptotic formulas match well with direct numerical simulations.

The good Boussinesq equation has several modified versions such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value is studied by Deift-Zhou nonlinear steepest descent analysis. In such initial condition, the asymptotic expressions of the Hirota-Satsuma equation and good Boussinesq equation away from the origin are derived and it is shown that the leading term of asymptotic formulas matches well the direct numerical simulations.

In this work, we propose a dual-port cell design to address the pass disturb in vertical NAND storage, which can pass signals through a dedicated and string-compatible pass gate. We demonstrate that: i) the pass disturb-free feature originates from weakening of the depolarization field by the pass bias at the high-${V}_{TH}$ (HVT) state and the screening of the applied field by channel at the low-${V}_{TH}$ (LVT) state; ii) combined simulations and experimental demonstrations of dual-port design verify the disturb-free operation in a NAND string, overcoming a key challenge in single-port designs; iii) the proposed design can be incorporated in a highly scaled vertical NAND FeFET string and the pass gate can be incorporated into the existing 3D NAND with the negligible overhead of the pass gate interconnection through a global bottom pass gate contact in the substrate., Comment: 29 pages, 7 figures

Both Sawada-Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are integrable systems with third-order Lax operator. Moreover, they are related with the same modified nonlinear equation (called modified SK-KK equation) by Miura transformations. This work first constructs the Riemann-Hilbert problem associated with the SK equation, KK equation and modified SK-KK equation by direct and inverse scattering transforms. Then the long-time asymptotics of these equations are studied based on Deift-Zhou steepest-descent method for Riemann-Hilbert problem. Finally, it is shown that the asymptotic solutions match very well with the results of direct numerical simulations., Comment: 50 pages,13 figures

The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work, the long-time asymptotics for the defocusing nonlinear Schr\"{o}dinger equation with general step-like initial data is investigated by the Whitham modulation theory and Riemann-Hilbert formulation. The Whitham modulation theory shows that there are six cases for the initial discontinuity problem according to the orders of the Riemann invariants. The leading-order terms and the corresponding error estimates for each region of the six cases are formulated by the Deift-Zhou nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. It is demonstrated that the long-time asymptotic solutions match very well with the results from Whitham modulation theory and the numerical simulations.

Single-port ferroelectric FET (FeFET) that performs write and read operations on the same electrical gate prevents its wide application in tunable analog electronics and suffers from read disturb, especially to the high-threshold voltage (VTH) state as the retention energy barrier is reduced by the applied read bias. To address both issues, we propose to adopt a read disturb-free dual-port FeFET where write is performed on the gate featuring a ferroelectric layer and the read is done on a separate gate featuring a non-ferroelectric dielectric. Combining the unique structure and the separate read gate, read disturb is eliminated as the applied field is aligned with polarization in the high-VTH state and thus improving its stability, while it is screened by the channel inversion charge and exerts no negative impact on the low-VTH state stability. Comprehensive theoretical and experimental validation have been performed on fully-depleted silicon-on-insulator (FDSOI) FeFETs integrated on 22 nm platform, which intrinsically has dual ports with its buried oxide layer acting as the non-ferroelectric dielectric. Novel applications that can exploit the proposed dual-port FeFET are proposed and experimentally demonstrated for the first time, including FPGA that harnesses its read disturb-free feature and tunable analog electronics (e.g., frequency tunable ring oscillator in this work) leveraging the separated write and read paths., Comment: 32 pages

In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -\Delta_{p}u+{\vert u\vert}^{p-2}u=f(u)-\mu u \quad \text{ in } \mathbb{R}^N,\\ {\Vert u\Vert}^2_{L^2(\mathbb{R}^N)}=m,\\ u\in W^{1,p}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N), \end{cases} \end{equation*} where $-\Delta_{p}u=div\left( {\vert\nabla u\vert}^{p-2}\nabla u \right)$, $2\leq p0$, $\mu\in\mathbb{R}$ will appear as a Lagrange multiplier and the continuous nonlinearity $f$ satisfies mass supercritical conditions. We mainly study the behavior of ground state energy $E_m$ with $m>0$ changing within a certain range and aim at extending nonlinear scalar field equation when $p=2$ and reducing the constraint condition of nonlinearity $f$., Comment: arXiv admin note: text overlap with arXiv:2002.03973 by other authors

In this work, we propose a ferroelectric FET(FeFET) time-domain compute-in-memory (TD-CiM) array as a homogeneous processing fabric for binary multiplication-accumulation (MAC) and content addressable memory (CAM). We demonstrate that: i) the XOR(XNOR)/AND logic function can be realized using a single cell composed of 2FeFETs connected in series; ii) a two-phase computation in an inverter chain with each stage featuring the XOR/AND cell to control the associated capacitor loading and the computation results of binary MAC and CAM are reflected in the chain output signal delay, illustrating full digital compatibility; iii) comprehensive theoretical and experimental validation of the proposed 2FeFET cell and inverter delay chains and their robustness against FeFET variation; iv) the homogeneous processing fabric is applied in hyperdimensional computing to show dynamic and fine-grain resource allocation to accommodate different tasks requiring varying demands over the binary MAC and CAM resources., Comment: 8 pages, 8 figures

Content addressable memory (CAM) is widely used in associative search tasks for its highly parallel pattern matching capability. To accommodate the increasingly complex and data-intensive pattern matching tasks, it is critical to keep improving the CAM density to enhance the performance and area efficiency. In this work, we demonstrate: i) a novel ultra-compact 1FeFET CAM design that enables parallel associative search and in-memory hamming distance calculation; ii) a multi-bit CAM for exact search using the same CAM cell; iii) compact device designs that integrate the series resistor current limiter into the intrinsic FeFET structure to turn the 1FeFET1R into an effective 1FeFET cell; iv) a successful 2-step search operation and a sufficient sensing margin of the proposed binary and multi-bit 1FeFET1R CAM array with sizes of practical interests in both experiments and simulations, given the existing unoptimized FeFET device variation; v) 89.9x speedup and 66.5x energy efficiency improvement over the state-of-the art alignment tools on GPU in accelerating genome pattern matching applications through the hyperdimensional computing paradigm., Comment: 20 pages, 14 figures

Camouflaging gate techniques are typically used in hardware security to prevent reverse engineering. Layout level camouflaging by adding dummy contacts ensures some level of protection against extracting the correct netlist. Threshold voltage manipulation for multi-functional logic with identical layouts has also been introduced for functional obfuscation. All these techniques are implemented at the expense of circuit-complexity and with significant area, energy, and delay penalty. In this paper, we propose an efficient hardware encryption technique with minimal complexity and overheads based on ferroelectric field-effect transistor (FeFET) active interconnects. The active interconnect provides run-time reconfigurable inverter-buffer logic by utilizing the threshold voltage programmability of the FeFETs. Our method utilizes only two FeFETs and an inverter to realize the masking function compared to recent reconfigurable logic gate implementations using several FeFETs and complex differential logic. We fabricate the proposed circuit and demonstrate the functionality. Judicious placement of the proposed logic in the IC makes it acts as a hardware encryption key and enables encoding and decoding of the functional output without affecting the critical path timing delay. Also, we achieve comparable encryption probability with a limited number of encryption units. In addition, we show a peripheral programming scheme for reconfigurable logic by reusing the existing scan chain logic, hence obviating the need for specialized programming logic and circuitry for keybit distribution. Our analysis shows an average encryption probability of 97.43% with an increase of 2.24%/ 3.67% delay for the most critical path/ sum of 100 critical paths delay for ISCAS85 benchmarks.

Deep random forest (DRF), which incorporates the core features of deep learning and random forest (RF), exhibits comparable classification accuracy, interpretability, and low memory and computational overhead when compared with deep neural networks (DNNs) in various information processing tasks for edge intelligence. However, the development of efficient hardware to accelerate DRF is lagging behind its DNN counterparts. The key for hardware acceleration of DRF lies in efficiently realizing the branch-split operation at decision nodes when traversing a decision tree. In this work, we propose to implement DRF through simple associative searches realized with ferroelectric analog content addressable memory (ACAM). Utilizing only two ferroelectric field effect transistors (FeFETs), the ultra-compact ACAM cell can perform a branch-split operation with an energy-efficient associative search by storing the decision boundaries as the analog polarization states in an FeFET. The DRF accelerator architecture and the corresponding mapping of the DRF model to the ACAM arrays are presented. The functionality, characteristics, and scalability of the FeFET ACAM based DRF and its robustness against FeFET device non-idealities are validated both in experiments and simulations. Evaluation results show that the FeFET ACAM DRF accelerator exhibits 10^6x/16x and 10^6x/2.5x improvements in terms of energy and latency when compared with other deep random forest hardware implementations on the state-of-the-art CPU/ReRAM, respectively., Comment: 44 pages, 16 figures

Nanoelectronic devices emulating neuro-synaptic functionalities through their intrinsic physics at low operating energies is imperative toward the realization of brain-like neuromorphic computers. In this work, we leverage the non-linear voltage dependent partial polarization switching of a ferroelectric field effect transistor to mimic plasticity characteristics of biological synapses. We provide experimental measurements of the synaptic characteristics for a $28nm$ high-k metal gate technology based device and develop an experimentally calibrated device model for large-scale system performance prediction. Decoupled read-write paths, ultra-low programming energies and the possibility of arranging such devices in a cross-point architecture demonstrate the synaptic efficacy of the device. Our hardware-algorithm co-design analysis reveals that the intrinsic plasticity of the ferroelectric devices has potential to enable unsupervised local learning in edge devices with limited training data.

The integrable focusing nonlinear Schrodinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the space-time plane where solutions display qualitatively distinct behaviors: an exponential-decay region, an algebraic-decay region, a non-oscillatory region, and an oscillatory region. Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, we compute the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions., Comment: Published version of the article. 42 pages, 13 figures

We investigate an integrable extended modified Korteweg-de Vries equation on the line with the initial value belonging to the Schwartz space. By performing the nonlinear steepest descent analysis of an associated matrix Riemann--Hilbert problem, we obtain the explicit leading-order asymptotics of the solution of this initial value problem as time $t$ goes to infinity. For a special case $\alpha=0$, we present the asymptotic formula of the solution to the extended modified Korteweg-de Vries equation in region $\mathcal{P}=\{(x,t)\in\bfR^2|0