Miura transformation for the “good” Boussinesq equation

It is well known that each solution of the modified Korteveg–de Vries (mKdV) equation gives rise, via the Miura transformation, to a solution of the Korteveg–de Vries (KdV) equation. In this work, we show that a similar Miura-type transformation exists also for the “good” Boussinesq equation. This transformation maps solutions of a second-order equation to solutions of the fourth-order Boussinesq equation. Just like in the case of mKdV and KdV, the correspondence exists also at the level of the underlying Riemann–Hilbert problems and this is in fact how we construct the new transformation.     

Simplex maximization of the correlation coefficient for DNA sizing analysis by capillary electrophoresis

The separation of DNA molecules in polymeric solution by capillary electrophoresis involves the optimization of several variables, such as polymer solution concentration, electric field separation, temperature, etc. The optimization of each variable individually usually is a time-consuming process and the results may reach a false optimum point. Chemometric methods are suitable to be applied in such cases in which a number of variables can be optimized simultaneously. The simplex is a chemometric method that can perform such a task easily and efficiently. In this study, a simplex method was carried out to maximize the correlation coefficient (r(2)) of a logarithmic plot of mobility (mu) vs. base pair (bp), which was obtained from the separation of DNA fragments of size between 75 and 4072 bp. The simplex showed three vertexes with r(2) > 0.98 and the vertex 21 showing the highest resolution. For the fragments between 201 and 2036 bp, the r(2) increased to 0.992 with and relative standard deviation (RSD) lower than 0.2% (inter- and intra-day variation). The precision of the method in determining the size of a PCR DNA fragment was carried out using a 1 kbp DNA ladder. With the addition of an internal standard to the sample, the precision could be further improved.     

Simulated moving bed technology with a simplified approach for protein purification. Separation of lactoperoxidase and lactoferrin from whey protein concentrate

Simulated moving bed (SMB) technology is a continuous chromatographic technique proven to have many advantages compared to conventional batch chromatography, such as: raised productivity and product concentration, reduced buffer consumption as well as more efficient use of raw material. In this study a 20 column SMB process for the separation of lactoperoxidase and lactoferrin from whey protein concentrate (WPC) was developed. A simplified approach with data from a single column experiment was used when designing the process. The SMB process data were compared to a theoretical scale-up of the breakthrough experiment reflecting the same 20 column set-up run in non-moving bed mode. The outcome of the comparison is a 48% raise in productivity, a 4.3 times decrease in buffer consumption, 6.5 times raise in target protein concentration with a raw material utilization which is slightly better for the SMB process.     

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

Dressing for a Novel Integrable Generalization of the Nonlinear Schrödinger Equation

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen.     

Efficient and reproducible analysis of peptides by capillary electrophoresis using noncovalently bilayer-coated capillaries

The usefulness of a noncovalent capillary coating consisting of two layers of oppositely charged polymers for the separation of peptides with capillary electrophoresis (CE) was studied. Capillaries were coated simply by subsequently flushing with solutions of 1% m/v Polybrene and 1% v/v poly(vinylsulfonate) (PVS) forming a bilayer, which showed to produce a strong and highly reproducible electroosmotic flow (EOF) at low pH. Using this coating in combination with a background electrolyte (BGE) containing sodium phosphate (pH 2.5) and 0.01% v/v PVS, initially broadened and overlapping peaks were obtained for some test peptides. By omitting the PVS from the BGE, the peak width and shape of the peptides improved resulting in baseline separation. A systematic study of the influence of the BGE composition showed that considerable further enhancement of the separation efficiency was achieved by increasing the ionic strength of the BGE. Using a BGE of 200 mM tris(hydroxymethyl)aminomethane (Tris)-phosphate (pH 2.5) plate numbers for the peptides were in the 300 000-600 000 range and the relative standard deviation of the peptide migration times was less then 0.3% (n = 5). The use of Tris-phosphate instead of sodium phosphate allowed the current to stay within acceptable limits when 30 kV was used as separation voltage. Overall, the bilayer coating showed a remarkable EOF repeatability, as well as long-term stability. Compared to bare fused-silica capillaries the intraday and interday repeatability of migration times was very favorable and coated capillaries could be used for over a month performing analyses with low and high ionic strength BGEs without any performance deterioration. The usefulness of the bilayer-coated capillaries for the analysis of positively charged peptides was demonstrated by the fast and efficient separation of various closely related enkephalins and the baseline separation of an isomeric peptide/peptoid couple exhibiting efficiencies of over 550 000 plates.     

Traveling wave solutions of the Degasperis-Procesi equation

We classify all weak traveling wave solutions of the Degasperis-Procesi equation. In addition to smooth and peaked solutions, the equation is shown to admit more exotic traveling waves such as cuspons, stumpons, and composite waves.     

Rigidity of complete hypersurfaces in the Euclidean space via anisotropic mean curvatures

In this paper, we study the rigidity of complete hypersurfaces in the Euclidean space Rⁿ⁺¹ via the so-called anisotropic mean curvatures, which correspond to a generalization of the usual mean curvatures. Under suitable constraints on the sign of the anisotropic mean curvatures and on the geometry of the height and angle functions naturally attached to such a hypersurface, we showthat it must be a hyperplane of Rⁿ⁺¹.     

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L²‐RH problems, we consider the construction of quarter plane solutions which are C¹ in time and C³ in space.