Numerical inverse scattering transform for the derivative nonlinear Schrodinger equation

In this paper, we develop the numerical inverse scattering transform (NIST) for solving the derivative nonlinear Schrodinger (DNLS) equation. The key technique involves formulating a Riemann-Hilbert problem (RHP) that is associated with the initial value problem and solving it numerically. Before solving the RHP, two essential operations need to be carried out. Firstly, high-precision numerical calculations are performed on the scattering data. Secondly, the RHP is deformed using the Deift-Zhou nonlinear steepest descent method. The DNLS equation has a continuous spectrum consisting of the real and imaginary axes and features three saddle points, which introduces complexity not encountered in previous NIST approaches. In our numerical inverse scattering method, we divide the $(x,t)$-plane into three regions and propose specific deformations for each region. These strategies not only help reduce computational costs but also minimize errors in the calculations. Unlike traditional numerical methods, the NIST does not rely on time-stepping to compute the solution. Instead, it directly solves the associated Riemann-Hilbert problem. This unique characteristic of the NIST eliminates convergence issues typically encountered in other numerical approaches and proves to be more effective, especially for long-time simulations

Suppression of blow-up in multi-species Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow in a finite channel

In this paper, we consider the multi-species parabolic-elliptic Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the 2-D Poiseuille flow $(\ A(1-y^2), 0\ )$ in a finite channel $\Omega=\mathbb{T}\times\mathbb{I}$ with $\mathbb{I}=(-1,1)$. Furthermore, the Navier-slip boundary condition is imposed on the perturbation of velocity $u$. We show that if the Poiseuille flow is sufficiently strong ($A$ is large enough), the solutions to the system are global in time without any smallness restriction on the initial cell mass

Efficient method for calculating the eigenvalue of the Zakharov-Shabat system

In this paper, a direct method is proposed to calculate the eigenvalue of the Zakharov-Shabat system. The main tools of our method are Chebyshev polynomials and the QR algorithm. After introducing the hyperbolic tangent mapping, the eigenfunctions and potential function defined in the real field can be represented by Chebyshev polynomials. Using Chebyshev nodes, the Zakharov-Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. The matrix eigenvalue problem is solved by the QR algorithm. Our method is used to calculate eigenvalues of the Zakharov-Shabat equation with three potentials, the rationality of our method is verified by comparison with analytical results

Effect of pH on CAHS D’s Secondary Structure Using FTIR Spectroscopy

Tardigrades are famous for their ability to survive extreme conditions such as complete desiccation. Cytosolic abundant heat soluble proteins (CAHS), a type of tardigrade disordered protein, is essential for desiccation survival. Our Lab has found that purified CAHS D undergo gelation, which I hypothesize is stabilized by intramolecular interactions involving transient formation of secondary structure. I investigated how pH impacts the secondary structure of CAHS D. Low (10 g/L) and high (40 g/L) concentration samples of CAHS D were prepared in buffer at pH 5.5 and 8.0. Their secondary structures were measured and compared by using Attenuated Total Internal Reflectance Fourier Transform Infrared (ATR-FTIR) spectroscopy. CAHS D’s N-terminal region contains histidine residues, whose pKa is between 5.5 and 8.0. At pH 8, 40 g/L CAHS D has 5±1 % more α-helix, 1±1% fewer turns and loops, 3.3±1.0% less random structures, and between 0.2±0.3% and 1.2±0.9% less β-sheet than at pH 5.5. Therefore, deprotonation of histidine increases the percentage of α-helix. Understanding the structure of CAHS D gels will aid in our understanding of its function

Effect of Trifluoroethanol on a Tardigrade Desiccation-Tolerance Protein

Protein-based drugs revolutionized medicine, yet require a cold-chain, low temperature transport and storage. Dry formulations offer a room temperature alternative. Tardigrades, a phylum of microscopic animals capable of surviving complete desiccation, offer a promising route towards this goal. My project focuses on a particular tardigrade desiccation-tolerance protein, cytosolic abundant heat soluble (CAHS) D. CAHS D protects client proteins from inactivation in vitro but the mechanism is unknown. My graduate student mentor and I showed that pure CAHS D forms a concentration-dependent thermoreversible gel cross-linked by intermolecular β-sheets, and we posit the gel matrix acts as a molecular shield upon desiccation. Here, trifluoroethanol (TFE) was used to mimic the effect of dehydration on CAHS D. Circular dichroism spectropolarimeter data indicated that low concentrations of CAHS D gained α-helix in TFE. At higher CAHS D concentrations, the protein went through liquid, gel, aggregate and phases, the latter with liquid-gel phase separation at increasing % TFE. Using attenuated total internal reflectance Fourier transformation infrared spectroscopy, I showed that gelation was due to intermolecular interactions between β-strands, which are not significant enough at low CAHS D concentration. I suggest that only at high CAHS D concentration did TFE mimic water deficiency, strengthening CAHS D gelation to let it act like a ‘molecular shield’ against water deficient environment.Bachelor of Scienc

A deep learning method for solving high-order nonlinear soliton equation

We propose effective scheme of deep learning method for high-order nonlinear soliton equation and compare the activation function for high-order soliton equation. The neural network approximates the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equation, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg de Vries equation. The results show that deep learning method can solve the high-order nonlinear soliton equation and reveal the interaction between solitons