scholarly journals Riemann-Hilbert Approach of The Complex Sharma-Tasso-Olver Equation and Its N-Soliton Solutions

2021 ◽  
Author(s):  
Sha Li ◽  
Tiecheng Xia ◽  
Jian Li
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Integrable Equation ◽  
The Matrix ◽  
Relationship Of ◽  
Parameter Values ◽  
The Relationship

Abstract In this paper, we use Riemann-Hilbert method to study the N-soliton solutions of the complex Sharma-Tasso-Olver(cSTO) equation. And then, based on analyzing the spectral problem of the Lax pair, the matrix Riemann-Hilbert problem for this integrable equation can be constructed, the N-soliton solutions about this system are given explicitly under the relationship of scattering matrix. At last, under the condition that some specifific parameter values are given, the three-dimensional diagram of the 2-soliton solution and the trajectory of the soliton solution will be simulated.

The N-soliton solutions to the Hirota and Maxwell–Bloch equation via the Riemann–Hilbert approach

2021 ◽  
pp. 2150153
Author(s):  
Jian Li ◽  
Tiecheng Xia ◽  
Hanyu Wei
Keyword(s):  
Physical Meaning ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bloch Equation ◽  
Lax Pair ◽  
Integrable Equation ◽  
Motion Trajectory ◽  
The Matrix ◽  
Riemann Hilbert Problem

In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

scholarly journals N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 826 ◽  
Author(s):  
Yuxin Lin ◽  
Huanhe Dong ◽  
Yong Fang
Keyword(s):  
Perturbation Theory ◽  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Nls Equation ◽  
Jump Matrix ◽  
Potential Matrix ◽  
Relationship Of ◽  
Riemann Hilbert Problem

In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of det P + and det P − ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R–H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R–H problem, the evolution equation of the R–H data with the perturbation is derived.

Riemann–Hilbert approach and N-soliton solutions for the Chen–Lee–Liu equation

2019 ◽  
Vol 33 (02) ◽  
pp. 1950002 ◽  
Author(s):  
Ming-Jun Xu ◽  
Tie-Cheng Xia ◽  
Bei-Bei Hu
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Potential Matrix ◽  
Riemann Hilbert Problem

In this paper, we construct the Riemann–Hilbert problem to the Lax pair of Chen–Lee–Liu (CLL) equation. As far as we know, many researchers have studied various equations with Riemann–Hilbert method before, but no one compared the N-soliton solutions calculated by different symmetries of potential matrix. Using different symmetries of potential matrix, we get two N-soliton solution formulae of the CLL equation. The interesting thing is that we find the equivalence of these two N-soliton solutions.

Riemann–Hilbert approach for the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation and its N-soliton solutions

2018 ◽  
Vol 32 (07) ◽  
pp. 1850088
Author(s):  
Hui Nie ◽  
Liping Lu ◽  
Xianguo Geng
Keyword(s):  
Spectral Analysis ◽  
Inverse Scattering ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Integrable Equation ◽  
Nonlinear Schrödinger ◽  
Nonlinear Integrable Equation ◽  
Riemann Hilbert Problem

On the basis of the spectral analysis for the Lax pair, a Riemann–Hilbert problem of the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation is established. Using the inverse scattering transformation and the Riemann–Hilbert approach, the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation is studied. As an application, N-soliton solutions of the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation are obtained. In addition, some figures are given to illustrate the soliton characteristics of the nonlinear integrable equation.

A Research on the Morphology and Composition of Flocs (Part 1: Floc Breaage Disregarded)

2013 ◽  
Vol 726-731 ◽  
pp. 1566-1572 ◽  
Author(s):  
Shi Qiang Ding ◽  
Qing Na Li ◽  
Xin Rong Pang ◽  
Ji Run Xu
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
New Classification ◽  
Suspension Viscosity ◽  
Primary Particles ◽  
Long Time ◽  
Relationship Of ◽  
The Relationship ◽  
Three Dimensional Space ◽  
Floc Breakage

The characteristics of flocs aggregated in flocculation have been paid more and more attention for a long time. In this paper, a new classification and analyses method dealing with the flocs is developed. The flocs formed after flocculation is divided into four kinds, including the left primary particles, linear flocs with all component particles in a line, planar flocs with all component particles on a plane and volumetric flocs with all component particles in a three-dimensional space. By analyzing the formation approaches of different kind of flocs regardless of the floc breakage, the number of every kind of floc is analyzed to be related with the suspension concentration mathematically. After comparing the different items in the models describing the relationship of floc number and concentration, a series of simplified expressions are presented. Lastly, a mathematical equation relating the measurable suspension viscosity with the numbers of different flocs is obtained.

scholarly journals Heart Rate Dependency and Inter-Lead Variability of the T Peak – T End Intervals

2020 ◽  
Vol 11 ◽  
Author(s):  
Irena Andršová ◽  
Katerina Hnatkova ◽  
Martina Šišáková ◽  
Ondřej Toman ◽  
Peter Smetana ◽  
...  
Keyword(s):  
Heart Rate ◽  
Three Dimensional ◽  
Rate Dependency ◽  
Rate Dependent ◽  
Heart Rate Correction ◽  
Ecg Leads ◽  
Relationship Of ◽  
The Relationship ◽  
Vector Magnitude

The electrocardiographic (ECG) assessment of the T peak–T end (Tpe) intervals has been used in many clinical studies, but several related physiological aspects have not been reported. Specifically, the sources of the Tpe differences between different ECG leads have not been systematically researched, the relationship of Tpe duration to underlying heart rate has not been firmly established, and little is known about the mutual correspondence of Tpe intervals measured in different ECG leads. This study evaluated 796,620 10-s 12-lead ECGs obtained from long-term Holters recorded in 639 healthy subjects (311 female) aged 33.8 ± 9.4 years. For each ECG, transformation to orthogonal XYZ lead was used to measure Tpe in the orthogonal vector magnitude (used as a reference for lead-to-lead comparisons) and to construct a three-dimensional T wave loop. The loop roundness was expressed by a ratio between its circumference and length. These ratios were significantly related to the standard deviation of Tpe durations in different ECG leads. At the underlying heart rate of 60 beats per minute, Tpe intervals were shorter in female than in male individuals (82.5 ± 5.6 vs 90.0 ± 6.5 ms, p < 0.0001). When studying linear slopes between Tpe intervals measured in different leads and the underlying heart rate, we found only minimal heart rate dependency, which was not systematic across the ECG leads and/or across the population. For any ECG lead, positive Tpe/RR slope was found in some subjects (e.g., 79 and 25% of subjects for V2 and V4 measurements, respectively) and a negative Tpe/RR slope in other subjects (e.g., 40 and 65% for V6 and V5, respectively). The steepest positive and negative Tpe/RR slopes were found for measurements in lead V2 and V4, respectively. In all leads, the Tpe/RR slope values were close to zero, indicating, on average, Tpe changes well below 2 ms for RR interval changes of 100 ms. On average, longest Tpe intervals were measured in lead V2, the shortest in lead III. The study concludes that the Tpe intervals measured in different leads cannot be combined. Irrespective of the measured ECG lead, the Tpe interval is not systematically heart rate dependent, and no heart rate correction should be used in clinical Tpe investigations.

scholarly journals Oculomotor activity parameters of the operator in the P300 brain computer interface and similar stimulus situations

2015 ◽  
Vol 8 (4) ◽  
pp. 134-144
Author(s):  
I.A. Basyul
Keyword(s):  
Average Length ◽  
Brain Computer Interface ◽  
Computer Interface ◽  
Similar Stimulus ◽  
The Matrix ◽  
Relationship Of ◽  
The Relationship ◽  
Visual Fixations ◽  
Oculomotor Activity ◽  
Stimulus Environment

Hypotheses about the relationship of the processes of visual perception and variations of the task in an identical stimulus environment was tested. The following tasks were tested: 1) a simple observation of the illuminations of the character in the matrix; 2) counting the number of highlights; 3) monitoring of the target symbol highlights and typing text with the P300 BCI. In a group of 14 people showed that the highest average length of visual fixation and the lowest dispersion of fixation observed for the second type of task. Statistically significant differences in the level of dispersion of visual fixations found between 1-2 and 1-3 modes; differences between the modes for the duration of fixations are at trends. Significant differences in the number of visual fixations on the target symbols wasn’t detected. The overall conclusion is the high perspective of pairing methodology brain-computer interface on the P300 wave with eye tracking to optimize the characteristics of the stimulus in the BCI environment. The differences in the parameters of oculomotor activity between the tasks reflect the level of attention concentration in the target symbols of the P300 BCI

Aerodynamic Characteristics Analysis of High-Speed Train on Cutting under Crosswinds

2013 ◽  
Vol 300-301 ◽  
pp. 62-67
Author(s):  
Kun Ye ◽  
Ren Xian Li
Keyword(s):  
High Speed ◽  
Three Dimensional ◽  
Aerodynamic Characteristics ◽  
High Speed Train ◽  
Aerodynamic Forces ◽  
Cutting Depth ◽  
Cutting Slope ◽  
Characteristics Analysis ◽  
Relationship Of ◽  
The Relationship

Cutting is an effective device to reduce crosswind loads acting on trains. The cutting depth, width and gradient of slope are important factors for design and construction of cutting. Based on numerical analysis methods of three-dimensional viscous incompressible aerodynamics equations, aerodynamic side forces and yawing moments acting on the high-speed train, with different depths and widths of cutting，are calculated and analyzed under crosswinds，meanwhile the relationship of the gradient of cutting slope and transverse aerodynamic forces acting on trains are also studied. Simulation results show that aerodynamic side forces and yawing moments acting on the train（the first, middle and rear train）decrease with the increase of cutting depth. The relationship between transverse forces (moments) coefficients acting on the three sections and the cutting depth basically is the three cubed relation. The bigger is cutting width，the worse is running stability of train. The relationship between yawing moments coefficients acting each body of the train and the cutting width approximately is the three cubed relation. The transverse Aerodynamic forces decreased gradually with the increase of the gradient of cutting slope, the relationship between yawing moments coefficients acting each body of the train and the gradient of cutting slope basically is the four cubed relation.

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