Bilinearization-reduction approach to integrable systems

Zhang Da-Jun

doi:10.7498/aps.72.20230063

Abstract

Authors and contacts

FullText HTML

1. 引　言
2. 实验信息
3. 实验结果与分析
3.1 辐照前后SiC MOSFET静态参数测试
3.2 栅特性测试
3.3 实验分析
4. 数值仿真
4.1 栅氧化层损伤仿真模拟
4.2 单粒子烧毁仿真模拟
5. 结　论

References

Cited By

Abstract

The paper is a review of the bilinearization-reduction method which provides an approach to obtain solutions to integrable systems. Many integrable coupled systems can be bilinearized and their solutions are presented in terms of double Wronskians (or double Casoratians in discrete case). The bilinearization-reduction method is based on bilinear equations and solutions in double Wronskian/Casoratian form. For those integrable equations that are reduced from coupled systems, one can first solve the unreduced coupled system, obtaining their solutions in double Wronskian/Casoratian form, then, implement suitable reduction techniques, so that solutions of the reduced equation can be obtained as reductions of those of the unreduced coupled system. The method proves effective in solving not only classical integrable equations but also the nonlocal ones. The so-called nonlocal integrable equations were introduced by Ablowitz and Musslimani via reductions with reverse-space (or reverse-time, or reverse-space-time). Note that this method particularly provides a convenient bilinear approach to solve nonlocal integrable systems. In this review, the nonlinear Schrödinger hierarchy and the differential-difference nonlinear Schrödinger equation are employed as demonstrative examples to elaborate this method. These two examples will be pedagogically helpful in understanding the reduction technique. The reduction is implemented by imposing suitable constraints on the basic column vectors of the double Wronskian/Casoratian. Realizations of the constraints are converted to solve a set of matrix equations which varies with the constraints. Special solutions of the matrix equations are provided, which are also helpful in understanding the eigenvalue structure of the involved spectral problems corresponding to the considered equations. Other examples include the Fokas-Lenells equation and the nonlinear Schrödinger equation with nontrivial background. Since many nonlinear equations with physical significance are integrable as reductions of integrable coupled systems, the paper provides a review as well as an introduction about the bilinearization-reduction method that can be used to solve these nonlinear integrable models.

Keywords:

PACS:

61.82.Fk	(Semiconductors)
71.55.Eq	(III-V semiconductors)
73.20.-r	(Electron states at surfaces and interfaces)
85.30.Tv	(Field effect devices)

Authors and contacts

Zhang Da-Jun

Department of Mathematics, Shanghai University, Shanghai 200444, China

Corresponding author: Zhang Da-Jun, djzhang@staff.shu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12271334, 12126352, 11875040)

FullText HTML

1. 引　言

作为第三代半导体材料, SiC材料具有禁带宽度大、击穿电场大、饱和漂移速度快和热导率高等优异性能, 可以用来制备抗辐射、大功率、高频和高温器件^[1–3]. 相比于Si基器件, 同等性能下, SiC基器件可以做到成本更低, 尺寸更小以及寿命更长, 因此SiC基器件更能满足新一代航天器对大功率、高频等性能方面的要求, 有望在空间和航天领域中扮演重要角色^[4–6]. 然而, SiC 金属-氧化物-半导体型场效应管(metal-oxide-semiconductor field-effect transistor, MOSFET)在空间领域的应用处于验证阶段, 由于空间环境中存在大量的高能重离子, SiC MOSFET经重离子辐照后会发生单粒子泄漏电流(single event leakage current, SELC)、单粒子栅穿(single event gate rupture, SEGR)或单粒子烧毁(single event burnout, SEB)^[7], 显著影响器件的工作.

近些年, 许多团队通过实验和仿真研究了重离子辐照下SiC MOSFET的退化和烧毁机制. Martinella等^[8,9]通过重离子辐照SiC MOSFET的实验发现在低偏置电压下, 被测器件的泄漏电流路径是从漏极到栅极, 涉及栅氧化层的损伤; 而在高偏置电压下, 泄漏电流路径主要是从漏极到源极, 这是由于碳化硅晶格的改变. Witulski等^[10]通过不同LET值的重离子实验发现器件SEB阈值电压会随入射离子LET值的增大而显著降低, 当LET > 10 MeV·cm²/mg时, SiC MOSFET的SEB阈值电压几乎为器件额定最大工作电压的一半. Ball等^[11]通过仿真模拟重离子入射SiC MOSFET器件, 发现离子入射诱导产生高局域态能量脉冲导致器件性能退化或者烧毁; 同时该团队^[12]通过实验对比和仿真模拟发现, 更厚的外延层、更低的掺杂浓度可以显著增大器件SEB阈值电压.

目前商用的SiC MOSFET分为平面栅型和沟槽型两种结构^[13], 在相同元胞尺寸下, 双沟槽型碳化硅MOSFET (double-trench MOSFET, DTMOSFET)相比于传统平面栅型碳化硅MOSFET(vertical double-diffused MOSFET, VDMOSFET), 具有更高的沟道迁移率、更低的比导通电阻以及更大的电流密度, 优异的性能使得SiC DTMOSFET具有更广阔的应用前景^[14–17]. 然而, 目前大多数研究都是针对SiC VDMOSFET的单粒子效应, 关于SiC DTMOSFET的单粒子效应研究较少. Zhou等^[18]通过TCAD模拟了不同LET值的重离子轰击SiC DTMOSFET器件敏感区, 研究其单粒子效应机理, 该团队^[19]还进行SiC DTMOSFET的2005 MeV Ta离子重离子实验, 实验结果表明其SEB阈值电压不超过其额定电压的42%. Cheng等^[20]借助TCAD模拟仿真软件研究了3种不同结构的槽栅型SiC MOSFET的单粒子效应, 发现DTMOSFET结构由于较深的源极深沟槽结构, 有助于快速收集离子入射后碰撞产生的载流子, 从而SEB阈值电压更高. 关于SiC DTMOSFET的单粒子效应研究缺乏更多的实验验证, 因此对SiC DTMOSFET的重离子实验有必要进一步开展和深入全面研究.

本文利用208 MeV Ge离子对碳化硅双沟槽MOSFET开展重离子辐照实验. 在辐照实验过程中, 实时监测了漏极电流和烧毁瞬间脉冲电流. 并在辐照实验后, 对器件进行了静态参数测试和栅特性测试. 结合TCAD工具, 模拟了不同偏置电压下重离子入射器件, 发现沿离子入射路径上的N^–外延层和N⁺衬底交界处温度超过了碳化硅的熔点, 这可能是导致器件发生SEB的原因.

2. 实验信息

本次实验选用的是Rohm公司生产的第四代双沟槽型碳化硅场效应晶体管SCT4060 KR, 其额定电流为26 A, 导通电阻为62 mΩ. 实验器件的单元结构如图1(a)所示, 在每个栅极沟槽的两个侧壁上都有沟道, 两侧都有一个源极沟槽, 延伸到漂移区, 更好地保护栅极氧化物. 在重离子实验前对器件进行了去封装处理, 以确保重离子能入射到器件的有源区, 并进行静态参数的测试, 以确保器件具有良好的电学性能.

$图 1 (a) SiC DTMOSFET单胞结构图; (b) 重离子实验电路原理图\r\nFig. 1. (a) SiC MOSFET cell structure; (b) schematic diagram of heavy ion experiment.$

图 1 (a) SiC DTMOSFET单胞结构图; (b) 重离子实验电路原理图

Fig. 1. (a) SiC MOSFET cell structure; (b) schematic diagram of heavy ion experiment.

下载: 全尺寸图片幻灯片

利用中国原子能科学研究院的HI-13串列加速器进行重离子实验, 选取能量为208 MeV的Ge离子, 其作用在器件表面的LET值为37.3 MeV·cm²/mg. 图1(b)为实验中使用的测试电路示意图, Keithley两台数字万用表分别监测和记录漏极电流和栅极电流, 示波器接入阻值为0.01 Ω的电阻两端, 用于捕获和存储器件烧毁瞬间的脉冲电流. 为了研究辐照损伤与源漏偏置电压的关系, 在辐照过程中, 器件的栅极电压V_g设置为0 V, 以确保器件处于关闭状态, 漏极电压V_DSirr设置为100, 200, 300, 400和500 V.

在重离子辐照实验中, 如果器件发生烧毁或者器件漏极电流持续增大, 注量累积到2×10⁶ ion/cm²时, 则停止辐照. 图2为不同偏置电压下SiC DTMOSFET的SELC和SEB. 如图2(a)所示, 当V_DSirr为100, 200, 300 V时, 漏极电流随重离子入射逐渐增大, 达到2×10⁶ ion/cm²的注量时, 电流未达到限制电流, 重离子入射到器件后产生的损伤导致漏极电流持续增长; 而在V_DSirr为400和500 V, 重离子入射到器件后在注量分别达到9×10⁴和3×10⁴ ion/cm²时, 漏极电流骤增到限制电流, 见图2(b), 器件发生单粒子烧毁, 器件源漏之间的电阻由数兆欧降到数百欧, 从而导致器件电流瞬间上升, 通过示波器可以捕获到这一电流的变化, 如图2(c)所示, V_DSirr为400 V时, 示波器捕获到的烧毁瞬间脉冲电流, 脉冲电流宽度为纳秒级别, 最大脉冲电流为4 A左右, 500 V同样出现此现象. 换而言之, 双沟槽SiC MOSFET的SEB阈值电压小于其额定电压的34%.

$图 2 辐照过程　(a) 漏极电流监测; (b) 烧毁电流监测; (c) 脉冲电流\r\nFig. 2. During irradiation: (a) Leakage current monitoring; (b) burn-out current monitoring; (c) pulse current monitoring.$

图 2 辐照过程　(a) 漏极电流监测; (b) 烧毁电流监测; (c) 脉冲电流

Fig. 2. During irradiation: (a) Leakage current monitoring; (b) burn-out current monitoring; (c) pulse current monitoring.

下载: 全尺寸图片幻灯片

3. 实验结果与分析

3.1 辐照前后SiC MOSFET静态参数测试

器件在辐照实验中表现出不同程度的退化, 这也导致其电学性能受到不同程度的损害. 图3给出了在辐照过程中的不同初始偏置电压下, 器件在辐照前后的传输特性曲线和输出特性曲线. 未烧毁器件以图3(a), (c)中辐照过程中初始偏置电压为100 V为例, 器件的阈值电压在辐照后出现了较小的负向偏移, 漂移了0.06 V, 然而, V_g = 6 V时的输出特性曲线显示漏极电流上升约77.51 mA; 其余偏压下变化见表1, 其退化程度与初偏压呈正相关. 烧毁器件以图3(b), (d)中辐照过程中偏置电压为400 V为例, 该器件在受到辐射后完全丧失其功能.

$图 3 辐照前后SiC MOSFET转移特征曲线和输出特征曲线\r\nFig. 3. Transfer characteristic curve and output characteristic curve of SiC MOSFET before and after irradiation.$

图 3 辐照前后SiC MOSFET转移特征曲线和输出特征曲线

Fig. 3. Transfer characteristic curve and output characteristic curve of SiC MOSFET before and after irradiation.

下载: 全尺寸图片幻灯片

表 1 辐照前后SiC MOSFET阈值电压偏移量和电流变化量

Table 1. Threshold voltage offset and current change of SiC MOSFET before and after irradiation.

辐照过程中初始偏置条件	ΔV_th/V	ΔI_d/mA
100 V	–0.06	77.51
200 V	–0.09	78.73
300 V	–0.16	80.56

下载: 导出CSV

| 显示表格

3.2 栅特性测试

进行辐照后栅特性测试. 栅压从0 V扫描到10 V, 源-漏短接, 测量栅极电流和漏极电流, 用于进一步确定辐照对器件造成损伤. 如图4所示, 辐照过程中初始偏置电压为100 V的被辐照器件泄漏电流与辐照前几乎无变化, 说明栅氧层并未受到损伤. 辐照过程中偏置电压大于200 V后, 被辐照的器件栅极泄漏电流的增长速度随着偏压的增大而增大, 说明其栅氧层的损伤也随之加深, 其泄漏电流退化程度与初始偏压呈正相关. 器件虽然没有发生SEB, 但因为泄漏电流的增大, 出于安全考虑也将不能在实际中正常使用.

$图 4 辐照后　(a)栅极泄漏电流特性; (b)漏极泄漏电流特性\r\nFig. 4. After irradiation: (a) Gate leakage currents characteristics; (b) drain leakage currents characteristics.$

图 4 辐照后　(a)栅极泄漏电流特性; (b)漏极泄漏电流特性

Fig. 4. After irradiation: (a) Gate leakage currents characteristics; (b) drain leakage currents characteristics.

下载: 全尺寸图片幻灯片

3.3 实验分析

通过双沟槽型碳化硅MOSFET重离子辐照前后静态参数的对比发现, 其阈值电压负漂, 泄漏电流增长. N沟道MOSFET的阈值电压 ${V}_{{\mathrm{t}}{\mathrm{h}}}$ 数学模型为^[21]

$\begin{split}{V}_{\text{th}}=\;&\frac{q}{{C}_{\text{OX}}}\left({N}_{\text{it}}-{N}_{\text{ot}}\right)+{C}_{\text{OX}}{\left(4{\varepsilon }_{\text{s}}q{N}_{\text{A}}{\phi }_{\text{Fp}}\right)}^{\frac{1}{2}}\\ &+2{\phi }_{\text{Fp}}+{\varphi }_{\text{MS}},\end{split}$

(1)

${\phi }_{\text{Fp}}=\frac{kT}{q}{\mathrm{ln}}\frac{{N}_{\text{A}}}{{n}_{\text{i}}},$

(2)

其中 ${C}_{{\mathrm{O}}{\mathrm{X}}}$ 为氧化层特征电容, ${N}_{{\mathrm{o}}{\mathrm{t}}}$ 为氧化层陷阱电荷面密度, ${N}_{{\mathrm{i}}{\mathrm{t}}}$ 为界面态陷阱电荷面密度, ${\varepsilon }_{{\mathrm{s}}}$ 为SiC的介电常, ${N}_{{\mathrm{A}}}$ 为P型基区的有效掺杂浓度, ${\phi }_{{\mathrm{F}}{\mathrm{p}}}$ 为准费米势, ${\varphi }_{{\mathrm{M}}{\mathrm{S}}}$ 为金属-半导体的功函数差, $k$ 为玻尔兹曼常数, $T$ 为绝对温度, ${n}_{{\mathrm{i}}}$ 为SiC的本征载流子浓度.

N沟道MOSFET的饱和区的电流数学模型为^[21]

${I}_{\text{DS}}=\frac{1}{2}{\mu }_{\text{n}}{C}_{\text{OX}}\frac{W}{L}{\left({V}_{\text{GS}}-{V}_{\text{th}}\right)}^{2},$

(3)

其中 ${\mu }_{{\mathrm{n}}}$ 为沟道载流子迁移率, $W$ 和 $L$ 分别为沟道的宽度和长度.

本文中, 阈值电压的负漂主要是由于重离子入射到器件在栅氧化层中产生电子-空穴对, 在栅氧化层中电子的迁移率明显高于空穴, 受到电场的作用, 电子快速向漏极移动, 而空穴缓慢移动, 在这个过程中会被栅氧化层的本征缺陷俘获, 导致栅氧化层陷阱电荷面密度 ${N}_{{\mathrm{o}}{\mathrm{t}}}$ 增大, 同时在SiC/SiO₂界面附近由重离子入射产生的空穴也可能在电场的作用下隧穿或者跃迁到氧化层被深能级缺陷俘获形成陷阱电荷, 最终阈值电压 ${V}_{{\mathrm{t}}{\mathrm{h}}}$ 降低^[22]. 由于阈值电压的负向漂移, 会导致器件SiC/SiO₂界面处的P型基区更容易形成反型层, 在较低的栅压下器件就能开启, 这解释了器件辐照后的输出特征曲线电流会出现了增长现象. 同时空穴在SiC/SiO₂界面处积累, 与源-漏偏压共同作用, 导致栅氧化层中的电场增大, 进而导致栅氧化层出现损伤, 导致栅应力测试中的栅电流增大, 其与源-漏偏压呈正相关^[19].

4. 数值仿真

根据图1(a)扫描电子显微镜的分析结果, 利用TCAD仿真工具建立了SiC DTMOSFET的二维模型, 其栅极沟槽的深度和宽度为1 μm和0.5 μm, 源极沟槽深度和宽度为1.8 μm和0.7 μm. 在进行二维结构仿真的过程中, 定性分析器件单粒子效应, 在模拟中添加了基本模型(包括和掺杂、温度相关的SRH复合模型、俄歇复合模型、迁移率模型和不完全电离模型等^[23]), 此外还添加了重离子入射模型和热力学模型. 其中重离子入射器件产生电子-空穴对的产生率^[24]可由公式(4)计算:

$G\left(l,w,t\right)={G}_{\text{LET}}\left(l\right)R\left(w,l\right)T\left(t\right),$

(4)

其中, $R\left(w, l\right)$ 和 $T\left(t\right)$ 是描述电子-空穴对产生率的时空分布函数, ${G}_{{\mathrm{L}}{\mathrm{E}}{\mathrm{T}}}\left(l\right)$ 是与入射粒子LET有关的LET产生密度, $w$ 和 $l$ 为入射粒子的轨迹半径和长度.

本文根据入射路径上经过不同区域为条件, 选择了5种不同入射路径, 如图5所示，入射点为路径 A经过多晶硅栅极中心、氧化层、外延层, 最终进入衬底; 路径 B经过多晶硅栅极边缘、氧化层、外延层, 最终进入衬底; 路径 C经过N⁺源区、P^–基区、外延层, 最终进入衬底; 路径 D经过N⁺源区、P^–屏蔽层、外延层, 最终进入衬底; 路径 E经过源极沟槽、P⁺屏蔽层、P^–屏蔽层、外延层, 最终进入衬底.

$图 5 离子入射位置示意图\r\nFig. 5. Diagram of ion strike positions.$

图 5 离子入射位置示意图

Fig. 5. Diagram of ion strike positions.

下载: 全尺寸图片幻灯片

4.1 栅氧化层损伤仿真模拟

图6为V_DSirr = 200 V, LET = 37.3 MeV·cm²/mg时, 不同位置入射器件栅氧化层电场强度随时间的演化过程. 仿真结果表明, 随着入射位置远离栅极中央, 栅氧化层中的电场强度逐渐变小. 由图7观察到, 离子从路径 A、路径 B和路径 C入射, 在栅氧化层拐角处都存在一个较大的电场强度, 并且从路径 B入射时, 其栅氧化层拐角处电场强度最大. 这是离子入射到器件中产生电子-空穴对, 空穴在栅氧化层的拐角处产生严重的堆积效应所导致. 较大的电场强度使器件的栅氧化层形成潜在损伤, 形成泄漏电流通道, 这种损伤机制解释了辐照后的器件在栅特性测试中栅极泄漏电流增大.

$图 6 VDSirr = 200 V时, 不同位置入射器件最大栅氧化层电场强度随时间的演化过程\r\nFig. 6. Evolution of maximum oxide electricfield at different strike positions with simulation time at VDSirr = 200 V.$

图 6 V_DSirr = 200 V时, 不同位置入射器件最大栅氧化层电场强度随时间的演化过程

Fig. 6. Evolution of maximum oxide electricfield at different strike positions with simulation time at V_DSirr = 200 V.

下载: 全尺寸图片幻灯片

$图 7 VDSirr = 200 V时, 重离子入射器件栅氧化层中电场强度分布图\r\nFig. 7. The distribution of electric field in gate oxide under heavy ion strike at VDSirr = 200 V.$

图 7 V_DSirr = 200 V时, 重离子入射器件栅氧化层中电场强度分布图

Fig. 7. The distribution of electric field in gate oxide under heavy ion strike at V_DSirr = 200 V.

下载: 全尺寸图片幻灯片

4.2 单粒子烧毁仿真模拟

碳化硅平面栅场效应晶体管的单粒子烧毁是由于其内部晶格温度高于其材料的熔点(3000 K)所导致^[25,26]. 因此在本文中, 将仿真模拟中器件内部晶格温度超过3000 K时所施加的源漏偏置电压定义为器件SEB的阈值电压.

从图8可以看到, 离子从路径 A入射到器件后不同时刻器件内部的晶格温度情况, 在重离子入射到器件后的N^–外延层和N⁺衬底处产生一个明显的高温点, 该点的晶格温度随着时间增大, 在t = 1 ns时, 达到最大值. 图9为沿离子入射路径的碰撞电离率和电场强度随深度的变化情况, 重离子入射器件后产生电子-空穴对, 电子在电场作用下向漏极移动, 使得该处电场强度进一步增大, 引起显著的碰撞电离, 在N^–外延层和N⁺衬底处碰撞电离率高达3.25×10²⁷ cm^–3·s^–1, 电离出更多的电子-空穴对, 电子在电场作用下继续被加速移动, 在局域产生大电流密度, 使得瞬时关态电流大大增大, 由热功率公式 $P = UI$ , 局域大电流密度导致其晶格温度超过SiC材料的熔点, 发生单粒子烧毁事件.

$图 8 重离子入射　(a) 1 ps, (b) 10 ps, (c) 100 ps和(d) 1 ns后器件内部晶格温度分布图\r\nFig. 8. The distribution of lattice temperature in device after heavy ion incident of (a) 1 ps, (b) 10 ps, (c) 100 ps, and (d) 1 ns.$

图 8 重离子入射　(a) 1 ps, (b) 10 ps, (c) 100 ps和(d) 1 ns后器件内部晶格温度分布图

Fig. 8. The distribution of lattice temperature in device after heavy ion incident of (a) 1 ps, (b) 10 ps, (c) 100 ps, and (d) 1 ns.

下载: 全尺寸图片幻灯片

$图 9 沿离子入射路径电场强度和碰撞电离率的分布情况\r\nFig. 9. Evolutions of impact ionization and electronic indensity along the ion track.$

图 9 沿离子入射路径电场强度和碰撞电离率的分布情况

Fig. 9. Evolutions of impact ionization and electronic indensity along the ion track.

下载: 全尺寸图片幻灯片

5. 结　论

本文研究了不同源漏偏置电压下双沟槽型碳化硅场效应晶体管的重离子单粒子效应. 实验结果表明, 与平面栅结构相比, 当辐照过程中偏置电压较低时, 双沟槽SiC MOSFET未出现泄漏电流通道, 当辐照过程中偏置电压大于200 V时, 辐照引起的泄漏电流在漏极到栅极和漏极到源极之间分配; 结合TCAD仿真模拟, 泄漏电流增大是由于栅氧化层拐角处高电场强度引起的损伤; 栅氧化层拐角处是沟槽型结构的敏感区域. 当LET值为37.3 MeV·cm²/mg, 双沟槽型碳化硅场效应晶体管的SEB阈值电压在400 V以下, 重离子入射到器件后, 在N^–外延层和N⁺衬底处的局域大电场强度和大电流密度导致晶格温度超过SiC的熔点, 引发单粒子烧毁事件. 这为宇航用碳化硅功率器件的辐照效应机理研究和应用提供了一定的参考和支撑.

References

[1]	Chen K, Deng X, Lou S Y, Zhang D J 2018 Stud. Appl. Math. 141 113 Google Scholar
[2]	Chen K, Zhang D J 2018 Appl. Math. Lett. 75 82 Google Scholar
[3]	Ablowitz M J, Musslimani Z H 2013 Phys. Rev. Lett. 110 064105
[4]	Ablowitz M J, Musslimani Z H 2014 Phys. Rev. E 90 032912
[5]	Ablowitz M J, Musslimani Z H 2021 Phys. Lett. A 409 127516
[6]	Deng X, Lou S Y, Zhang D J 2018 Appl. Math. Comput. 332 477 Google Scholar
[7]	Chen K, Na C N, Yang J X 2023 Nonlinear Dyn. 111 1685 Google Scholar
[8]	Feng W, Zhao S L, Sun Y Y 2020 Int. J. Mod. Phys. B 34 2050021
[9]	Silem A, Wu H, Zhang D J 2021 Appl. Math. Lett. 116 107049
[10]	Chen K, Liu S M, Zhang D J 2019 Appl. Math. Lett. 88 230
[11]	Wang J, Wu H, Zhang D J 2020 Commun. Theor. Phys. 72 045002
[12]	Shi Y, Shen S F, Zhao S L 2019 Nonlinear Dyn. 95 1257 Google Scholar
[13]	Liu S Z, Wu H 2021 Mod. Phys. Lett. B 35 2150410
[14]	Wang J, Wu H 2022 Nonlinear Dyn. 109 3101 Google Scholar
[15]	Liu S M, Wu H, Zhang D J 2020 Rep. Math. Phys. 86 271 Google Scholar
[16]	Liu S Z, Wang J, Zhang D J 2022 Stud. Appl. Math. 148 651 Google Scholar
[17]	Wu H 2021 Nonlinear Dyn. 106 2497 Google Scholar
[18]	Liu S M, Wang J, Zhang D J 2022 Rep. Math. Phys. 89 199 Google Scholar
[19]	Wang J, Wu H, Zhang D J 2022 Chin. Phys. B 31 120201
[20]	Wang J, Wu H 2022 Commun. Nonlinear Sci. Numer. Simul. 104 106052
[21]	Zhang D J, Liu S M, Deng D 2023 Open Commun. Nonlinear Math. Phys. 3 23
[22]	Zhang D J 2020 Wronskian solutions of integrable systems, in Nonlinear Systems and Their Remarkable Mathematical Structures (Vol. 2) (Eds. Euler N, Nucci M C) (Boca Raton: CRC Press, Taylor & Francis) pp415–444
[23]	Hirota R 1974 Prog. Theore. Phys. 52 1498 Google Scholar
[24]	Freeman N C, Nimmo J J C 1983 Phys. Lett. A 95 1 Google Scholar
[25]	Nimmo J J C 1983 Phys. Lett. A 99 279 Google Scholar
[26]	Hietarinta J, Zhang D J 2009 J. Phys. A: Math. Theor. 42 404006
[27]	Ablowitz M J, Kaup D J, Newell A C, Segur H 1973 Phys. Rev. Lett. 31 125 Google Scholar
[28]	陈登远 2006 孤子引论 (北京: 科学出版社) Chen D Y 2006 Introduction to Soliton Theory (Beijing: Science Press) (in Chinese)
[29]	Newell A C 1985 Solitons in Mathematics and Physics (Philadelphin: SIAM)
[30]	Liu Q M 1990 J. Phys. Soc. Jpn. 59 3520 Google Scholar
[31]	Yin F M, Sun Y P, Cai F Q, Chen D Y 2008 Comm. Theore. Phys. 49 401 Google Scholar
[32]	Ablowitz M J, Ladik J F 1976 J. Math. Phys. 17 1011 Google Scholar
[33]	Fokas A S 1995 Physica D 87 145 Google Scholar
[34]	Lenells J, Fokas A S 2009 Nonlinearity 22 11 Google Scholar
[35]	Lenells J 2009 Stud. Appl. Math. 123 215 Google Scholar
[36]	Gerdjikov V S, Ivanov M I, Kulish P P 1980 Theor. Math. Phys. 44 784 Google Scholar
[37]	Zhang D J 2006 arXiv: nlin/0603008v3 [nlin.SI]
[38]	Zhang D J, Zhao S L, Sun Y Y, Zhou J 2014 Rev. Math. Phys. 26 1430006
[39]	Gürses M, Pekcan A 2018 J. Math. Phys. 59 051501
[40]	Gürses M, Pekcan A 2019 Commun. Nonlinear Sci. Numer. Simul. 71 161 Google Scholar
[41]	Gürses M, Pekcan A 2021 Commun. Nonlinear Sci. Numer. Simul. 97 105736

Cited By

期刊类型引用(1)

岑政，向飞，吴兆希，杨蕊亦，叶思楠，王祖正. 基于LabVIEW的SiC MOSFET器件单粒子效应测试系统. 电子元器件与信息技术. 2024(09): 26-29 .

百度学术

其他类型引用(1)

表 1 矩阵方程(26)的解

Table 1. Solutions to matrix equation (26)

No.	$(\sigma, \delta)$	${\boldsymbol T}$	${\boldsymbol A}$
1)	$(1, -1)$	$T_1=T_4=({\bf{0}})_N$ , $T_3=-T_2 ={\boldsymbol{I} }_{N}$	$K_1=-K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
2)	$(1, 1)$	$T_1=T_4=({\bf{0}})_N$ , $T_3=T_2 ={\boldsymbol{I} }_{N}$	$K_1=-K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
3)	$(-1, -1)$	$T_1=T_4=({\bf{0}})_N$ , $T_3=T_2 ={\boldsymbol{I} }_{N}$	$K_1=K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
4)	$(-1, 1)$	$T_1=T_4=({\bf{0}})_N$ , $T_3=-T_2 ={\boldsymbol{I} }_{N}$	$K_1=K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
5)	$(-1, -1)$	$T_1=-T_4= {\boldsymbol{I} }_{N}, \; T_2=T_3 =({\bf{0} })_{N}$	$K_1={\boldsymbol{K} }_{N}\in \mathbb{R}_{N\times N}, \; K_4=-{\boldsymbol{H} }_{N}\in \mathbb{R}_{N\times N}$

DownLoad: CSV

[1]	Chen K, Deng X, Lou S Y, Zhang D J 2018 Stud. Appl. Math. 141 113 Google Scholar
[2]	Chen K, Zhang D J 2018 Appl. Math. Lett. 75 82 Google Scholar
[3]	Ablowitz M J, Musslimani Z H 2013 Phys. Rev. Lett. 110 064105
[4]	Ablowitz M J, Musslimani Z H 2014 Phys. Rev. E 90 032912
[5]	Ablowitz M J, Musslimani Z H 2021 Phys. Lett. A 409 127516
[6]	Deng X, Lou S Y, Zhang D J 2018 Appl. Math. Comput. 332 477 Google Scholar
[7]	Chen K, Na C N, Yang J X 2023 Nonlinear Dyn. 111 1685 Google Scholar
[8]	Feng W, Zhao S L, Sun Y Y 2020 Int. J. Mod. Phys. B 34 2050021
[9]	Silem A, Wu H, Zhang D J 2021 Appl. Math. Lett. 116 107049
[10]	Chen K, Liu S M, Zhang D J 2019 Appl. Math. Lett. 88 230
[11]	Wang J, Wu H, Zhang D J 2020 Commun. Theor. Phys. 72 045002
[12]	Shi Y, Shen S F, Zhao S L 2019 Nonlinear Dyn. 95 1257 Google Scholar
[13]	Liu S Z, Wu H 2021 Mod. Phys. Lett. B 35 2150410
[14]	Wang J, Wu H 2022 Nonlinear Dyn. 109 3101 Google Scholar
[15]	Liu S M, Wu H, Zhang D J 2020 Rep. Math. Phys. 86 271 Google Scholar
[16]	Liu S Z, Wang J, Zhang D J 2022 Stud. Appl. Math. 148 651 Google Scholar
[17]	Wu H 2021 Nonlinear Dyn. 106 2497 Google Scholar
[18]	Liu S M, Wang J, Zhang D J 2022 Rep. Math. Phys. 89 199 Google Scholar
[19]	Wang J, Wu H, Zhang D J 2022 Chin. Phys. B 31 120201
[20]	Wang J, Wu H 2022 Commun. Nonlinear Sci. Numer. Simul. 104 106052
[21]	Zhang D J, Liu S M, Deng D 2023 Open Commun. Nonlinear Math. Phys. 3 23
[22]	Zhang D J 2020 Wronskian solutions of integrable systems, in Nonlinear Systems and Their Remarkable Mathematical Structures (Vol. 2) (Eds. Euler N, Nucci M C) (Boca Raton: CRC Press, Taylor & Francis) pp415–444
[23]	Hirota R 1974 Prog. Theore. Phys. 52 1498 Google Scholar
[24]	Freeman N C, Nimmo J J C 1983 Phys. Lett. A 95 1 Google Scholar
[25]	Nimmo J J C 1983 Phys. Lett. A 99 279 Google Scholar
[26]	Hietarinta J, Zhang D J 2009 J. Phys. A: Math. Theor. 42 404006
[27]	Ablowitz M J, Kaup D J, Newell A C, Segur H 1973 Phys. Rev. Lett. 31 125 Google Scholar
[28]	陈登远 2006 孤子引论 (北京: 科学出版社) Chen D Y 2006 Introduction to Soliton Theory (Beijing: Science Press) (in Chinese)
[29]	Newell A C 1985 Solitons in Mathematics and Physics (Philadelphin: SIAM)
[30]	Liu Q M 1990 J. Phys. Soc. Jpn. 59 3520 Google Scholar
[31]	Yin F M, Sun Y P, Cai F Q, Chen D Y 2008 Comm. Theore. Phys. 49 401 Google Scholar
[32]	Ablowitz M J, Ladik J F 1976 J. Math. Phys. 17 1011 Google Scholar
[33]	Fokas A S 1995 Physica D 87 145 Google Scholar
[34]	Lenells J, Fokas A S 2009 Nonlinearity 22 11 Google Scholar
[35]	Lenells J 2009 Stud. Appl. Math. 123 215 Google Scholar
[36]	Gerdjikov V S, Ivanov M I, Kulish P P 1980 Theor. Math. Phys. 44 784 Google Scholar
[37]	Zhang D J 2006 arXiv: nlin/0603008v3 [nlin.SI]
[38]	Zhang D J, Zhao S L, Sun Y Y, Zhou J 2014 Rev. Math. Phys. 26 1430006
[39]	Gürses M, Pekcan A 2018 J. Math. Phys. 59 051501
[40]	Gürses M, Pekcan A 2019 Commun. Nonlinear Sci. Numer. Simul. 71 161 Google Scholar
[41]	Gürses M, Pekcan A 2021 Commun. Nonlinear Sci. Numer. Simul. 97 105736

[1]	WANG Gangwei, TAN Zixuan. Similarity transformations and exact solutions of the generalized (n + 1)-dimensional Schrödinger equation with (2m + 1)-th order nonlinear terms and spatiotemporally varying coeffcients. Acta Physica Sinica, 2025, 74(10): . doi: 10.7498/aps.74.20250225
[2]	. From Transverse Field Ising Chain to Quantum E8 Integrable Model. Acta Physica Sinica, 2022, (): . doi: 10.7498/aps.71.20211836
[3]	Wang Xiao, Yang Jia-Hao, Wu Jian-Da. From the transverse field Ising chain to the quantum E₈ integrable model. Acta Physica Sinica, 2021, 70(23): 230504. doi: 10.7498/aps.70.20211836
[4]	Xu Dan-Hong, Lou Sen-Yue. Dark soliton molecules in nonlinear optics. Acta Physica Sinica, 2020, 69(1): 014208. doi: 10.7498/aps.69.20191347
[5]	Lou Sen-Yue. Full reversal symmetric multiple soliton solutions for integrable systems. Acta Physica Sinica, 2020, 69(1): 010503. doi: 10.7498/aps.69.20191172
[6]	Zhang Li-Xiang, Liu Han-Ze, Xin Xiang-Peng. Symmetry reductions, exact equations and the conservation laws of the generalized (3+1) dimensional Zakharov-Kuznetsov equation. Acta Physica Sinica, 2017, 66(8): 080201. doi: 10.7498/aps.66.080201
[7]	Li Kai-Hui, Liu Han-Ze, Xin Xiang-Peng. Lie symmetry analysis, optimal system, exact solutions and conservation laws of a class of high-order nonlinear wave equations. Acta Physica Sinica, 2016, 65(14): 140201. doi: 10.7498/aps.65.140201
[8]	Liu Yong, Liu Xi-Qiang. Exact solutions of Whitham-Broer-Kaup equations with variable coefficients. Acta Physica Sinica, 2014, 63(20): 200203. doi: 10.7498/aps.63.200203
[9]	Wan Hui. Exact solutions to the nonlinear diffusion-convection equation with variable coefficients and source term. Acta Physica Sinica, 2013, 62(9): 090203. doi: 10.7498/aps.62.090203
[10]	Li Ning, Liu Xi-Qiang. Symmetries, reductions and exact solutions of Broer-Kau-Kupershmidt system. Acta Physica Sinica, 2013, 62(16): 160203. doi: 10.7498/aps.62.160203
[11]	Zeng Wen-Li, Ma Song-Hua, Ren Qing-Bao. Exact solutions and soliton excitations for the (2+1)-dimensional Bogoyavlenskii-Schiff system. Acta Physica Sinica, 2012, 61(11): 110508. doi: 10.7498/aps.61.110508
[12]	Li Bang-Qing, Ma Yu-Lan. (G′/G)-expansion method and new exact solutions for (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov system. Acta Physica Sinica, 2009, 58(7): 4373-4378. doi: 10.7498/aps.58.4373
[13]	Taogetusang, Sirendaoerji. A method for constructing exact solutions of nonlinear evolution equation with variable coefficients. Acta Physica Sinica, 2009, 58(4): 2121-2126. doi: 10.7498/aps.58.2121
[14]	Li Bo, Wang Yan-Shen. The q-boson hopping model with integrable open boundary condition. Acta Physica Sinica, 2007, 56(3): 1260-1265. doi: 10.7498/aps.56.1260
[15]	Han Zhao-Xiu. New exact solutions for nonlinear Klein-Gordon equations. Acta Physica Sinica, 2005, 54(4): 1481-1484. doi: 10.7498/aps.54.1481
[16]	Huang Ding-Jiang, Zhang Hong-Qing. Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations. Acta Physica Sinica, 2004, 53(8): 2434-2438. doi: 10.7498/aps.53.2434
[17]	Zhang Yu-Feng, Yan Qing-You. A type of expanding integrable system for NLS-mKdV hierarchy. Acta Physica Sinica, 2003, 52(9): 2109-2113. doi: 10.7498/aps.52.2109
[18]	Li De-Sheng, Zhang Hong-Qing. The new doubly-periodic solutions for nonlinear coupled scalar field equations( Ⅱ). Acta Physica Sinica, 2003, 52(10): 2379-2385. doi: 10.7498/aps.52.2379
[19]	Li De-Sheng, Zhang Hong-Qing. The new doubly-periodic solutions for nonlinear coupled scalar field equations( Ⅰ). Acta Physica Sinica, 2003, 52(10): 2373-2378. doi: 10.7498/aps.52.2373
[20]	Zhu Hong-Yi, Shen Jian-Qi. . Acta Physica Sinica, 2002, 51(7): 1448-1452. doi: 10.7498/aps.51.1448

期刊类型引用(1)
1. 岑政，向飞，吴兆希，杨蕊亦，叶思楠，王祖正. 基于LabVIEW的SiC MOSFET器件单粒子效应测试系统. 电子元器件与信息技术. 2024(09): 26-29 . 百度学术
其他类型引用(1)

Catalog

Vol. 72, Issue 10 - 2023-05-20

Metrics

Abstract views: 5966
PDF Downloads: 212
Cited By: 2

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板