The elliptic sine-Gordon equation in a half plane

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case. We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation. We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.

The Klein-Gordon equation in a domain with time-dependent boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time-dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.

The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.

Spectral analysis of the elliptic sine-Gordon equation in the quarter plane

We study the elliptic sine-Gordon equation in the quarter plane using a spectral transform approach. We determine the Riemann-Hilbert problem associated with well-posed boundary value problems in this domain and use it to derive a formal representation of the solution. Our analysis is based on a generalization of the usual inverse scattering transform recently introduced by Fokas for studying linear elliptic problems.

The Dirichlet-to-Neumann map for the elliptic sine Gordon

We analyse the Dirichlet problem for the elliptic sine Gordon equation in the upper half plane. We express the solution $q(x,y)$ in terms of a Riemann-Hilbert problem whose jump matrix is uniquely defined by a certain function $b(\la)$, $\la\in\R$, explicitly expressed in terms of the given Dirichlet data $g_0(x)=q(x,0)$ and the unknown Neumann boundary value $g_1(x)=q_y(x,0)$, where $g_0(x)$ and $g_1(x)$ are related via the global relation $\{b(\la)=0$, $\la\geq 0\}$. Furthermore, we show that the latter relation can be used to characterise the Dirichlet to Neumann map, i.e. to express $g_1(x)$ in terms of $g_0(x)$. It appears that this provides the first case that such a map is explicitly characterised for a nonlinear integrable {\em elliptic} PDE, as opposed to an {\em evolution} PDE.

Boundary value problems for the elliptic sine-Gordon equation in a semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem whose \jump matrix" depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q_y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h. The determination of the function h remains open.

Contingency locus in ctsR of Listeria monocytogenes Scott A: a strategy for occurrence of abundant piezotolerant isolates within clonal populations

In a recent study we demonstrated that a high-hydrostatic-pressure-tolerant isolate of Listeria monocytogenes lacks a codon in the class 3 heat shock regulator gene ctsR. This mutation in the region that encodes four consecutive glycines was directly responsible for the observed piezotolerance, increased stress resistance, and reduced virulence. The aim of the present study was to determine whether mutations in ctsR are frequently associated with piezotolerance in L. monocytogenes. Wild-type cultures of L. monocytogenes were therefore exposed to 350 MPa for 20 min, and the piezotolerance of individual surviving isolates was assessed. This rendered 33 isolates with a stable piezotolerant phenotype from a total of 84 survivors. Stable piezotolerant mutants were estimated to be present in the initial wild-type population at frequencies of >10�5. Subsequent sequencing of the ctsR gene of all stable piezotolerant isolates revealed that two-thirds of the strains (i.e., n � 21) had mutations in this gene. The majority of the mutations (16 of 21 strains) consisted of a triplet deletion in the glycine-encoding region of ctsR, identical to what was found in our previous study. Interestingly, 2 of 21 mutants contained a codon insertion in this repeat region. The remaining three stable piezotolerant strains showed a 19-bp insertion in the glycine repeat region, a 16-bp insertion downstream of the glycine repeat area (both leading to frameshifts and a truncated ctsR), and an in-frame 114-bp deletion encoding a drastically shortened carboxy terminus of CtsR. In four instances it was not possible to generate a PCR product. A piezotolerant phenotype could not be linked to mutations in ctsR in 8 of 33 isolates, indicating that other thus-far-unknown mechanisms also lead to stable piezotolerance. The present study highlights the importance of ctsR in piezotolerance and stress tolerance of L. monocytogenes, and it demonstrates that short-sequence repeat regions contribute significantly to the occurrence of a piezotolerant and stress-tolerant subpopulation within L. monocytogenes cultures, thus playing an important role in survival.

Characterization of a Listeria monocytogenes Scott A isolate with high tolerance towards high hydrostatic pressure

An isolate of L. monocytogenes Scott A that is tolerant to high hydrostatic pressure (HHP), named AK01, was isolated upon a single pressurization treatment of 400 MPa for 20 min and was further characterized. The survival of exponential- and stationary-phase cells of AK01 in ACES [N-(2-acetamido)-2-aminoethanesulfonic acid] buffer was at least 2 log units higher than that of the wild type over a broad range of pressures (150 to 500 MPa), while both strains showed higher HHP tolerance (piezotolerance) in the stationary than in the exponential phase of growth. In semiskim milk, exponential-phase cells of both strains showed lower reductions upon pressurization than in buffer, but again, AK01 was more piezotolerant than the wild type. The piezotolerance of AK01 was retained for at least 40 generations in rich medium, suggesting a stable phenotype. Interestingly, cells of AK01 lacked flagella, were elongated, and showed slightly lower maximum specific growth rates than the wild type at 8, 22, and 30°C. Moreover, the piezotolerant strain AK01 showed increased resistance to heat, acid, and H2O2 compared with the wild type. The difference in HHP tolerance between the piezotolerant strain and the wild-type strain could not be attributed to differences in membrane fluidity, since strain AK01 and the wild type had identical in situ lipid melting curves as determined by Fourier transform infrared spectroscopy. The demonstrated occurrence of a piezotolerant isolate of L. monocytogenes underscores the need to further investigate the mechanisms underlying HHP resistance of food-borne microorganisms, which in turn will contribute to the appropriate design of safe, accurate, and feasible HHP treatments.

The combined action of carvacrol and high hydrostatic pressure on Listeria monocytogenes Scott A

Aims: The aim of the study was to investigate the combined antimicrobial action of the plantderived volatile carvacrol and high hydrostatic pressure (HHP). Methods and Results: Combined treatments of carvacrol and HHP have been studied at different temperatures, using exponentially growing cells of Listeria monocytogenes, and showed a synergistic action. The antimicrobial effects were higher at 1°C than at 8 or 20°C. Furthermore, addition of carvacrol to cells exposed to sublethal HHP treatment caused similar reductions in viable numbers as simultaneous treatment with carvacrol and HHP. Synergism was also observed between carvacrol and HHP in semi-skimmed milk that was artifcially contaminated with L. monocytogenes. Conclusions: Carvacrol and HHP act synergistically and the antimicrobial effects of the combined treatment are greater at lower temperatures. Significance and Impact of the Study: The study demonstrates the synergistic antimicrobial effect of essential oils in combination with HHP and indicates the potential of these combined treatments in food processing.