Linear integral equations, infinite matrices, and soliton hierarchies

A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all (2 + 1)- and (1 + 1)-dimensional soliton hierarchies associated with scalar differential spectral problems. The integrability characteristics for the obtained soliton hierarchies, including Miura-type transforms, τ-functions, Lax pairs, and soliton solutions, are also derived within this framework

A varA variational principle for discrete integrable systemsiational principle for discrete integrable systems

For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference equations with two independent variables, MDC allows us to define an action on arbitrary 2-dimensional surfaces embedded in a higher dimensional space of independent variables, where the action is not only a functional of the field variables but also the choice of surface. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also with respect to variations of the surface in the space of independent variables. Here we derive the resulting system of generalized Euler-Lagrange equations arising from that principle. We treat the case where the equations are 2 dimensional (but which due to MDC can be consistently embedded in higher-dimensional space), and show that they can be integrated to yield relations of quadrilateral type. We also derive the extended set of Euler-Lagrange equations for 3-dimensional systems, i.e., those for equations with 3 independent variables. The emerging point of view from this study is that the variational principle can be considered as the set of equations not only encoding the equations of motion but as the defining equations for the Lagrangians themselves

Suicide among persons with childhood leukaemia in Slovenia

Pri osebah, ki so v otroštvu zbolele za rakom, so pogosto prisotne telesne in psihosocialne posledice bolezni ter njenega zdravljenja. Mnoge raziskave so pokazale, da je pri osebah z izkušnjo raka v otroštvu depresivnost in samomorilno vedenje močneje izraženo. V naši raziskavi smo proučili pojavljanje samomorov pri osebah, ki so v otroštvu zbolele za levkemijo, v primerjavi s splošno populacijo v Sloveniji, v obdobju 1978–2010. Pričakovano število samomorov smo izračunali na osnovi kontrolne skupine posameznikov iz splošne populacije, ki je bila s skupino preiskovancev, tj. oseb, ki so v otroštvu zbolele za levkemijo, izenačena po spolu, starosti ob začetku opazovanja, letu začetka opazovanja in dolžini opazovanja. Raziskava je pokazala, da med tistimi, ki so v otroštvu zboleli za levkemijo, v letih 1978–2010 nobena oseba ni storila samomora, kar se statistično značilno ne razlikuje od pričakovanega števila samomorov (0,448) v primerljivi splošni populaciji v Sloveniji. Ugotovitve raziskave nakazujejo, da kljub znano bolj izraženem samomorilnem vedenju med preživelimi raka v otroštvu v Sloveniji v primerjavi s splošno populacijo pojavljanje samomorov pri osebah, zbolelih za levkemijo v otroštvu, ni pogostejše kot v splošni populaciji.Persons with childhood leukaemia often suffer from physical and psychosocial consequences of the disease and its treatment. Several studies have shown that depression and suicidal behaviour are expressed strongly in persons with a childhood cancer experience. In our study, we researched the occurrence of suicides among persons with childhood leukaemia compared to the general population in Slovenia in the period 1978–2010. The expected number of suicides was calculated based on the control group of individuals from the general population with the same gender, age at the beginning of observation, starting year and duration of observation as the research group, thus group of persons with childhood cancer. The study showed that none of the persons with childhood cancer committed suicide in the period 1978-2010, which is not statistically different from the expected number of suicides (0.448) in comparison with the general population in Slovenia. The findings of this study indicate that, despite the significantly increased expression of suicidal behaviour among survivors of childhood leukaemia in Slovenia compared to the general population, suicides do not occur more often among people with childhood leukaemia than among the general population

The discrete potential Boussinesq equation and its multisoliton solutions

An alternate form of discrete potential Boussinesq equation is proposed and its multisoliton solutions are constructed. An ultradiscrete potential Boussinesq equation is also obtained from the discrete potential Boussinesq equation using the ultradiscretization technique. The detail of the multisoliton solutions is discussed by using the reduction technique. The lattice potential Boussinesq equation derived by Nijhoff et al. is also investigated by using the singularity confinement test. The relation between the proposed alternate discrete potential Boussinesq equation and the lattice potential Boussinesq equation by Nijhoff et al. is clarified.Comment: 17 pages,To appear in Applicable Analysis, Special Issue of Continuous and Discrete Integrable System

Algebro-geometric integration of the Q1 lattice equation via nonlinear integrable symplectic maps

The Q1 lattice equation, a member in the Adler–Bobenko–Suris list of 3D consistent lattices, is investigated. By using the multidimensional consistency, a novel Lax pair for Q1 equation is given, which can be nonlinearized to produce integrable symplectic maps. Consequently, a Riemann theta function expression for the discrete potential is derived with the help of the Baker–Akhiezer functions. This expression leads to the algebro-geometric integration of the Q1 lattice equation, based on the commutativity of discrete phase flows generated from the iteration of integrable symplectic maps

Closed-form modified Hamiltonians for integrable numerical integration schemes

Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually diverges. In contrast, this paper constructs and analyzes explicit examples of nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. We present cases of one- and twodegrees symplectic mappings arising as reductions of nonlinear integrable lattice equations, for which the modified Hamiltonians can be computed in closed form. These modified Hamiltonians are also given as power series in the time step by Yoshida’s method based on the Baker–Campbell–Hausdorff series. Another example displays an implicit dependence on the time step which could be of relevance to certain implicit schemes in numerical analysis. In light of these examples, the potential importance of integrable mappings to the field of geometric numerical integration is discussed

Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations

A unified framework is presented for the solution structure of three-dimensional discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearizing transform, which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained

Lagrangian multiforms for Kadomtsev-Petviashvili (KP) and the Gelfand-Dickey hierarchy

We present, for the first time, a Lagrangian multiform for the complete Kadomtsev-Petviashvili (KP) hierarchy -- a single variational object that generates the whole hierarchy and encapsulates its integrability. By performing a reduction on this Lagrangian multiform, we also obtain Lagrangian multiforms for the Gelfand-Dickey hierarchy of hierarchies, comprising, amongst others, the Korteweg-de Vries and Boussinesq hierarchies