N-soliton solutions to the DKP equation and Weyl group actions

We study soliton solutions to the DKP equation which is defined by the Hirota bilinear form, {\begin{array}{llll} (-4D_xD_t+D_x^4+3D_y^2) \tau_n\cdot\tau_n=24\tau_{n-1}\tau_{n+1}, (2D_t+D_x^3\mp 3D_xD_y) \tau_{n\pm 1}\cdot\tau_n=0 \end{array} \quad n=1,2,.... where $\tau_0=1$. The $\tau$-functions $\tau_n$ are given by the pfaffians of certain skew-symmetric matrix. We identify one-soliton solution as an element of the Weyl group of D-type, and discuss a general structure of the interaction patterns among the solitons. Soliton solutions are characterized by $4N\times 4N$ skew-symmetric constant matrix which we call the $B$-matrices. We then find that one can have $M$-soliton solutions with $M$ being any number from $N$ to $2N-1$ for some of the $4N\times 4N$ $B$-matrices having only $2N$ nonzero entries in the upper triangular part (the number of solitons obtained from those $B$-matrices was previously expected to be just $N$).Comment: 22 pages, 12 figure

On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy

We describe the interaction pattern in the $x$-$y$ plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, $(-4u_{t}+u_{xxx}+6uu_x)_{x}+3u_{yy}=0$. Those solutions also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for $y\to \pm\infty$, and we show that all the solutions (except the one-soliton solution) are of {\it resonant} type, consisting of arbitrary numbers of line solitons in both aymptotics; that is, arbitrary $N_-$ incoming solitons for $y\to -\infty$ interact to form arbitrary $N_+$ outgoing solitons for $y\to\infty$. We also discuss the interaction process of those solitons, and show that the resonant interaction creates a {\it web-like} structure having $(N_--1)(N_+-1)$ holes.Comment: 18 pages, 16 figures, submitted to JPA; Math. Ge

Solitons in the Higgs phase -- the moduli matrix approach --

We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge theory with N_F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Moduli spaces of domain walls (kinks) and vortices, which are the only elementary solitons in the Higgs phase, are found in terms of the moduli matrix. Stable monopoles and instantons can exist in the Higgs phase if they are attached by vortices to form composite solitons. The moduli spaces of these composite solitons are also worked out in terms of the moduli matrix. Webs of walls can also be formed with characteristic difference between Abelian and non-Abelian gauge theories. We characterize the total moduli space of these elementary as well as composite solitons. Effective Lagrangians are constructed on walls and vortices in a compact form. We also present several new results on interactions of various solitons, such as monopoles, vortices, and walls. Review parts contain our works on domain walls (hep-th/0404198, hep-th/0405194, hep-th/0412024, hep-th/0503033, hep-th/0505136), vortices (hep-th/0511088, hep-th/0601181), domain wall webs (hep-th/0506135, hep-th/0508241, hep-th/0509127), monopole-vortex-wall systems (hep-th/0405129, hep-th/0501207), instanton-vortex systems (hep-th/0412048), effective Lagrangian on walls and vortices (hep-th/0602289), classification of BPS equations (hep-th/0506257), and Skyrmions (hep-th/0508130).Comment: 89 pages, 33 figures, invited review article to Journal of Physics A: Mathematical and General, v3: typos corrected, references added, the published versio

Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation ($P_{II}$). Here we define two-variable polynomials $Y_{n}(t,h)$ on a lattice with spacing $h$, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce when $h=0$. They also provide rational solutions for a particular discretisation of $P_{II}$, namely the so called {\it alternate discrete} $P_{II}$, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation ($P_{III}$). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete $P_{II}$, which recovers Jimbo and Miwa's Lax pair for $P_{II}$ in the continuum limit $h\to 0$.Comment: 23 pages, IOP style. Title changed, and connection with Umemura polynomials adde

Multi-indexed Wilson and Askey-Wilson Polynomials

As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials. They are obtained from the original Wilson and Askey-Wilson polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.Comment: 30 pages. Three references added. To appear in J.Phys.A. arXiv admin note: text overlap with arXiv:1203.586

Pathophysiology and pathogenesis of circadian rhythm sleep disorders

Metabolic, physiological and behavioral processes exhibit 24-hour rhythms in most organisms, including humans. These rhythms are driven by a system of self-sustained clocks and are entrained by environmental cues such as light-dark cycles as well as food intake. In mammals, the circadian clock system is hierarchically organized such that the master clock in the suprachiasmatic nuclei of the hypothalamus integrates environmental information and synchronizes the phase of oscillators in peripheral tissues. The transcription and translation feedback loops of multiple clock genes are involved in the molecular mechanism of the circadian system. Disturbed circadian rhythms are known to be closely related to many diseases, including sleep disorders. Advanced sleep phase type, delayed sleep phase type and nonentrained type of circadian rhythm sleep disorders (CRSDs) are thought to result from disorganization of the circadian system. Evaluation of circadian phenotypes is indispensable to understanding the pathophysiology of CRSD. It is laborious and costly to assess an individual's circadian properties precisely, however, because the subject is usually required to stay in a laboratory environment free from external cues and masking effects for a minimum of several weeks. More convenient measurements of circadian rhythms are therefore needed to reduce patients' burden. In this review, we discuss the pathophysiology and pathogenesis of CRSD as well as surrogate measurements for assessing an individual's circadian phenotype

Internal Ribosomal Entry Site-Mediated Translation Is Important for Rhythmic PERIOD1 Expression

The mouse PERIOD1 (mPER1) plays an important role in the maintenance of circadian rhythm. Translation of mPer1 is directed by both a cap-dependent process and cap-independent translation mediated by an internal ribosomal entry site (IRES) in the 5′ untranslated region (UTR). Here, we compared mPer1 IRES activity with other cellular IRESs. We also found critical region in mPer1 5′UTR for heterogeneous nuclear ribonucleoprotein Q (HNRNPQ) binding. Deletion of HNRNPQ binding region markedly decreased IRES activity and disrupted rhythmicity. A mathematical model also suggests that rhythmic IRES-dependent translation is a key process in mPER1 oscillation. The IRES-mediated translation of mPer1 will help define the post-transcriptional regulation of the core clock genes

The Functional Interplay between Protein Kinase CK2 and CCA1 Transcriptional Activity Is Essential for Clock Temperature Compensation in Arabidopsis

Circadian rhythms are daily biological oscillations driven by an endogenous mechanism known as circadian clock. The protein kinase CK2 is one of the few clock components that is evolutionary conserved among different taxonomic groups. CK2 regulates the stability and nuclear localization of essential clock proteins in mammals, fungi, and insects. Two CK2 regulatory subunits, CKB3 and CKB4, have been also linked with the Arabidopsis thaliana circadian system. However, the biological relevance and the precise mechanisms of CK2 function within the plant clockwork are not known. By using ChIP and Double–ChIP experiments together with in vivo luminescence assays at different temperatures, we were able to identify a temperature-dependent function for CK2 modulating circadian period length. Our study uncovers a previously unpredicted mechanism for CK2 antagonizing the key clock regulator CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1). CK2 activity does not alter protein accumulation or subcellular localization but interferes with CCA1 binding affinity to the promoters of the oscillator genes. High temperatures enhance the CCA1 binding activity, which is precisely counterbalanced by the CK2 opposing function. Altering this balance by over-expression, mutation, or pharmacological inhibition affects the temperature compensation profile, providing a mechanism by which plants regulate circadian period at changing temperatures. Therefore, our study establishes a new model demonstrating that two opposing and temperature-dependent activities (CCA1-CK2) are essential for clock temperature compensation in Arabidopsis

High-Throughput Chemical Screen Identifies a Novel Potent Modulator of Cellular Circadian Rhythms and Reveals CKIα as a Clock Regulatory Kinase

A novel compound “longdaysin” was found to dramatically slow down the speed of the circadian clock through simultaneous inhibition of protein kinases CKIδ, CKIα, and ERK2

Emergence of Noise-Induced Oscillations in the Central Circadian Pacemaker

Computational modeling and experimentation explain how intercellular coupling and intracellular noise can generate oscillations in a mammalian neuronal network even in the absence of cell-autonomous oscillators