Miura Type Research Articles

Simplifications of Lax pairs for differential–difference equations by gauge transformations and (doubly) modified integrable equations

Matrix differential–difference Lax pairs play an essential role in the theory of integrable nonlinear differential–difference equations. We present sufficient conditions which allow one to simplify such a Lax pair by matrix gauge transformations. Furthermore, we describe a procedure for such a simplification and present applications of it to constructing new integrable equations connected by (non-invertible) discrete substitutions of Miura type to known equations with Lax pairs.Suppose that one has three (possibly multicomponent) equations E, E1, E2, a (Miura-type) discrete substitution from E1 to E, and a discrete substitution from E2 to E1. Then E1 and E2 can be called a modified version of E and a doubly modified version of E, respectively. We demonstrate how the above-mentioned procedure helps (in the considered examples) to construct modified and doubly modified versions of a given equation possessing a Lax pair satisfying certain conditions.The considered examples include scalar equations of Itoh–Narita–Bogoyavlensky type and 2-component equations related to the Toda lattice. We present several new integrable equations connected by new discrete substitutions of Miura type to known equations.

Miura Type Transform Between Non-Abelian Volterra and Toda Lattices and Inverse Spectral Problem for Band Operators

Miura Type Transform Between Non-Abelian Volterra and Toda Lattices and Inverse Spectral Problem for Band Operators

Rh(III)-Catalyzed Weakly Coordinating 2-Pyridone-Directed Oxidative Annulation Using Internal Alkynes: A Reversal in Selectivity.

A rhodium(III)-catalyzed Satoh-Miura type oxidative annulation of N-aryl 2-pyridone derivatives is described using internal alkyne as a coupling partner. A weakly coordinating carbonyl group of the 2-pyridone ring is utilized for this transformation. The reaction proceeds with a broad scope and wide functional group tolerance. The solvent plays an important role in the developed method to furnish a different class of annulated product. A preliminary investigation was carried out to explore the photophysical properties of the obtained polyarylated N-naphthyl 2-pyridones.

Novel NPR2 Gene Mutations Affect Chondrocytes Function via ER Stress in Short Stature.

Natriuretic peptide receptor 2 (NPR2) plays a key role in cartilage and bone morphogenesis. The NPR2 gene mutations result in acromesomelic dysplasia, Maroteaux type (AMDM), short stature with nonspecific skeletal abnormalities (SNSK), and epiphyseal chondrodysplasia, Miura type (ECDM). However, the pathogenic mechanism remains unclear. In our study, we identified one de novo (R557C) and six novel variants (G602W, V970F, R767*, R363*, F857S, and Y306S) in five independent Chinese families with familial short stature. Three patients with heterozygous mutations (G602W, V970F, and R767*) were diagnosed with SNSK (height SD score ranged from −2.25 to −5.60), while another two with compound heterozygous mutations (R363* and F857S, R557C and Y306S) were diagnosed with AMDM (height SD score ranged from −3.10 to −5.35). Among three patients with heterozygous status, two patients before puberty initiation with rhGH treatment significantly improved their growth (height velocity 7.2 cm/year, 6.0 cm/year), and one patient in puberty had a poor response to the rhGH treatment (height velocity 2.5 cm/year). Seven NPR2 gene variants were constructed and overexpressed in HEK293T and ATDC5 cells, and we found that ATDC5 cells with mutant NPR2 gene showed decreased differentiation, as evidenced by lower expression of ColII, ColX, and BMP4 and higher expression of Sox9. Moreover, the apoptosis rate was elevated in ATDC5 cells expressing the mutant NPR2 gene. N-glycosylation modification, plasma membrane localization, and ER stress resulted from the accumulation of mutant protein in ER, as shown by the higher expression of GRP78 and p-IRE1α. Overall, our results provide a novel insight into NPR2 loss of function, which could promote chondrocyte apoptosis and repress cell differentiation through ER stress and the unfolded protein response.

A new family with epiphyseal chondrodysplasia type Miura.

Epiphyseal chondrodysplasia, Miura type (ECDM) is a skeletal dysplasia with tall stature and distinctive skeletal features caused by heterozygous NPR2 pathogenic variants. Only four families have been reported. We present a family with five affected individuals (mother, three sons, and daughter). The mother's phenotype was relatively mild: borderline tall stature and elongated halluces operated during childhood. The children were remarkably more severely affected with tall stature, scoliosis, and elongated toes and fingers leading to suspicion of Marfan syndrome. Progressive valgus deformities (at the hips, knees, and ankles) were the main complaints and necessitated orthopedic investigations and surgery. Radiographs showed coxa valga, scoliosis, multiple pseudoepiphyses of the fingers and toes with uneven elongation of the digits and ankle valgus. The two older brothers underwent osteotomies and guided growth for axial deformities and arthrodesis for elongated halluces. Genetic testing confirmed the clinical diagnosis of ECDM: all affected individuals had a heterozygous c.2647G>A (p.Val883Met) NPR2 variant in a highly conserved region in the carboxyl-terminal guanylyl cyclase domain. This two-generation family elucidates the clinical and radiological variability of the disease. These rare cases are important to gain further understanding of the fundamental processes of growth regulation.

Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation

We consider five-point differential-difference equations. Our aim is to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated to transformations of the first, second and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable five-point equations and new transformations have been found. Moreover, we have derived one new completely discrete equation. There are a few non-standard transformations which are of the Miura type or are linearizable in a non-standard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.

CREB activation in hypertrophic chondrocytes is involved in the skeletal overgrowth in epiphyseal chondrodysplasia Miura type caused by activating mutations of natriuretic peptide receptor B.

Natriuretic peptide receptor B (NPRB) produces cyclic guanosine monophosphate (cGMP) when bound by C-type natriuretic peptide (CNP). Activating mutations in NPRB cause a skeletal overgrowth disorder, which has been named epiphyseal chondrodysplasia, Miura type (ECDM; OMIM #615923). Here we explored the cellular and molecular mechanisms for the skeletal overgrowth in ECDM using a mouse model in which an activating mutant NPRB is specifically expressed in chondrocytes. The mutant mice (NPRB[p.V883M]-Tg) exhibited postnatal skeletal overgrowth and increased cGMP in cartilage. Both endogenous and transgene-derived NPRB proteins were localized at the plasma membrane of hypertrophic chondrocytes. The hypertrophic zone of growth plate was thickened in NPRB[p.V883M]-Tg. An in vivo BrdU-labeling assay suggested that some of the hypertrophic chondrocytes in NPRB[p.V883M]-Tg mice continued to proliferate, although wild-type (WT) chondrocytes stopped proliferating after they became hypertrophic. In vitro cell studies revealed that NPRB activation increased the phosphorylation of cyclic AMP-responsive element binding protein (CREB) and expression of cyclin D1 in matured chondrocytes. Treatment with cell-permeable cGMP also enhanced the CREB phosphorylation. Inhibition of cyclic adenosine monophosphate (cAMP)/protein kinase A pathway had no effects on the CREB phosphorylation induced by NPRB activation. In immunostaining of the growth plates for the proliferation marker Ki67, phosphorylated CREB and cyclin D1, most signals were similarly observed in the proliferating zone in both genotypes, but some cells in the hypertrophic zone of NPRB[p.V883M]-Tg were also positively stained. These results suggest that NPRB activation evokes its signal in hypertrophic chondrocytes to induce CREB phosphorylation and make them continue to proliferate, leading to the skeletal overgrowth in ECDM.

Integrable Möbius-invariant evolutionary lattices of second order

We solve the classification problem for integrable lattices of the form $u_{,t}=f(u_{-2},\dots,u_2)$ under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains 5 equations, including 3 new. Difference Miura type substitutions are found which relate these equations with known polynomial lattices. We also present some classification results for the generic lattices.

Non-invertible transformations of differential–difference equations

We discuss aspects of the theory of non-invertible transformations of differential–difference equations and, in particular, the notion of Miura type transformation. We introduce the concept of non-Miura type linearizable transformation and we present techniques that allow one to construct simple linearizable transformations and might help one to solve classification problems. This theory is illustrated by the example of a new integrable differential–difference equation depending on five lattice points, interesting from the viewpoint of the non-invertible transformation, which relate it to an Itoh–Narita–Bogoyavlensky equation.

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.

Some Special Rogue Waves in One Type of Variable Coefficient Nonlinear Schrödinger Equations

In this paper, a similarity transformation is devised to construct a Miura type transformations between one type of variable coefficient nonlinear Schrodinger equations and the cubic NLS equation. This transformation is devoted to obtain the travelling wave solutions of a complicated equation by using the solutions of a simpler equation directly. The result shows different rogue wave solutions of the variable coefficient nonlinear Schrodinger equations are given easily.

Infinite-dimensional prolongation Lie algebras and multicomponent Landau–Lifshitz systems associated with higher genus curves

The Wahlquist–Estabrook prolongation method constructs for some PDEs a Lie algebra that is responsible for Lax pairs and Bäcklund transformations of certain type. We present some general properties of Wahlquist–Estabrook algebras for (1+1)-dimensional evolution PDEs and compute this algebra for the n-component Landau–Lifshitz system of Golubchik and Sokolov for any n≥3. We prove that the resulting algebra is isomorphic to the direct sum of a 2-dimensional abelian Lie algebra and an infinite-dimensional Lie algebra L(n) of certain matrix-valued functions on an algebraic curve of genus 1+(n−3)2n−2. This curve was used by Golubchik, Sokolov, Skrypnyk, and Holod in constructions of Lax pairs. Also, we find a presentation for the algebra L(n) in terms of a finite number of generators and relations. These results help to obtain a partial answer to the problem of classification of multicomponent Landau–Lifshitz systems with respect to Bäcklund transformations.Furthermore, we construct a family of integrable evolution PDEs that are connected with the n-component Landau–Lifshitz system by Miura type transformations parametrized by the above-mentioned curve. Some solutions of these PDEs are described.

New Miura type transformations between integrable dispersive wave equations

In this paper, an algebraic method is devised to construct new Miura type transformations between integrable dispersive wave equations. The characteristic feature of our method lies in that the travelling wave solutions of an aimed equation can be determined by the solutions of a simpler equation directly. Our work is an attempt in searching for travelling solutions of complicate nonlinear equations.

Transformations of integrable hydrodynamic chains and their hydrodynamic reductions

Hydrodynamic reductions of the hydrodynamic chain associated with the dispersionless limit of the 2+1-dimensional Harry-Dym equation are found by using the Miura type and reciprocal transformations applied to the Benney hydrodynamic chain.

Gauge transformations and symmetries of integrable systems

We analyse several integrable systems in zero-curvature form within the framework of invariant gauge theory. In the Drinfeld–Sokolov gauge, we derive a two-parameter family of nonlinear evolution equations which as special cases include the Korteweg–de Vries (KdV) and Harry Dym (HD) equations. We find residual gauge transformations which lead to infinitesimal symmetries of this family of equations. For KdV and HD equations we find an infinite hierarchy of such symmetry transformations, and we investigate their relation with local conservation laws, constants of the motion and the bi-Hamiltonian structure of the equations. Applying successive gauge transformations of Miura type we obtain a sequence of gauge-equivalent integrable systems, among them the modified KdV and Calogero KdV equations.

Miura type transformations and homogeneous spaces

We relate Miura type transformations (MTs) over an evolution system to its zero-curvature representations with values in Lie algebras g. We prove that certain homogeneous spaces of g produce MTs and show how to distinguish these spaces. For a scalar translation-invariant evolution equation this allows to classify all MTs in terms of homogeneous spaces of the Wahlquist-Estabrook algebra of the equation. For other evolution systems this allows to construct some MTs. As an example, we study MTs over the KdV equation, a 5th order equation of Harry-Dym type, and the coupled KdV-mKdV system of Kersten and Krasilshchik.

The Gurevich–Zybin system

We present three different linearizable extensions of the Gurevich-Zybin system. Their general solutions are found by reciprocal transformations. In this paper we rewrite the Gurevich-Zybin system as a Monge-Ampere equation. By application of reciprocal transformation this equation is linearized. Infinitely many local Hamiltonian structures, local Lagrangian representations, local conservation laws and local commuting flows are found. Moreover, all commuting flows can be written as Monge-Ampere equations similar to the Gurevich-Zybin system. The Gurevich-Zybin system describes the formation of a large scale structures in the Universe. The second harmonic wave generation is known in nonlinear optics. In this paper we prove that the Gurevich-Zybin system is equivalent to a degenerate case of the second harmonic generation. Thus, the Gurevich-Zybin system is recognized as a degenerate first negative flow of two-component Harry Dym hierarchy up to two Miura type transformations. A reciprocal transformation between the Gurevich-Zybin system and degenerate case of the second harmonic generation system is found. A new solution for the second harmonic generation is presented in implicit form.

Integrable Equations Arising from Motions of Plane Curves. II

Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity, and projective geometries on the real line, and restricted conformal, conformal, and projective geometries in the plane. Together with Chou and Qu [Physica D 162 (2002), 9–33], we determine inextensible motions and their associated integrable equations in all Klein geometries in the plane. The relations between several pairs of these geometries provide a natural geometric explanation of the existence of transformations of Miura and Cole-Hopf type.

Higher Dimensional Integrable Models from Lower Ones via Miura Type Deformation Relation

Abstract To find nontrivial high dimensional integrable models (especially in (3+1)-dimensions) is one of the most important problems in nonlinear physics. A systematic method to find some nontrivial high dimensional integrable models is established by means of noninvertible deformation relations. Starting from a noninvertible Miura type transformation, we find that the (1+1)-dimensional sinh-Gordon model appearing in many physical fields is a deformation of the (0+1)-dimensional Riccati equation. A high dimensional Miura type deformation (including two different (3+1)-dimensional reductions) of the heat conduction equation is proved to be Painlevé integrable. Some special types of explicit exact solutions, like multi-plane and/or multi-camber soliton solutions, multi-dromion solutions and multiple ring soliton solutions, are obtained.

Fully supersymmetric hierarchies from an energy-dependent super Hill operator

A super Hill operator with energy dependent potentials is proposed and the associated integrable hierarchy is constructed explicitly. It is shown that in the general case, the resulted hierarchy is multi-Hamiltonian system. The Miura type transformations and modified hierarchies are also presented.