Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows \map(t,x) in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarchies utilizes a moving parallel frame and connection 1-form along the curve flows, related to the Klein geometry of the Lie group $G\supset H$ where $H$ is the local frame structure group. The soliton equations arise in explicit form from the induced flow on the frame components of the principal normal vector N=\covder{x}\mapder{x} along each curve, and display invariance under the equivalence subgroup in $H$ that preserves the unit tangent vector T=\mapder{x} in the framing at any point $x$ on a curve. Their bi-Hamiltonian integrability structure is shown to be geometrically encoded in the Cartan structure equations for torsion and curvature of the parallel frame and its connection 1-form in the tangent space T_\map M of the curve flow. The hierarchies include group-invariant versions of sine-Gordon (SG) and modified Korteweg-de Vries (mKdV) soliton equations that are found to be universally given by curve flows describing non-stretching wave maps and mKdV analogs of non-stretching Schrodinger maps on $G/H$. These results provide a geometric interpretation and explicit bi-Hamiltonian formulation for many known multicomponent soliton equations. Moreover, all examples of group-invariant (multicomponent) soliton equations given by the present geometric framework can be constructed in an explicit fashion based on Cartan's classification of symmetric spaces.Comment: Published version, with a clarification to Theorem 4.5 and a correction to the Hamiltonian flow in Proposition 5.1

Hamiltonian evolutions of twisted gons in \RP^n

In this paper we describe a well-chosen discrete moving frame and their associated invariants along projective polygons in \RP^n, and we use them to write explicit general expressions for invariant evolutions of projective $N$-gons. We then use a reduction process inspired by a discrete Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the space of projective invariants, and we establish a close relationship between the projective $N$-gon evolutions and the Hamiltonian evolutions on the invariants of the flow. We prove that {any} Hamiltonian evolution is induced on invariants by an evolution of $N$-gons - what we call a projective realization - and we give the direct connection. Finally, in the planar case we provide completely integrable evolutions (the Boussinesq lattice related to the lattice $W_3$-algebra), their projective realizations and their Hamiltonian pencil. We generalize both structures to $n$-dimensions and we prove that they are Poisson. We define explicitly the $n$-dimensional generalization of the planar evolution (the discretization of the $W_n$-algebra) and prove that it is completely integrable, providing also its projective realization

Symplectically-invariant soliton equations from non-stretching geometric curve flows

A moving frame formulation of geometric non-stretching flows of curves in the Riemannian symmetric spaces $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ is used to derive two bi-Hamiltonian hierarchies of symplectically-invariant soliton equations. As main results, multi-component versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting $Sp(1)\times Sp(n-1)$ invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schr\"odinger map.Comment: 39 pages; remarks added on algebraic aspects of the moving frame used in the constructio

The blackgrass genome reveals patterns of non-parallel evolution of polygenic herbicide resistance

Globally, weedy plants are a major constraint to sustainable crop production. Much of the success of weeds rests with their ability to rapidly adapt in the face of human-mediated management of agroecosystems. Alopecurus myosuroides (blackgrass) is a widespread and impactful weed affecting agriculture in Europe. Here we report a chromosome-scale genome assembly of blackgrass and use this reference genome to explore the genomic/genetic basis of non-target site herbicide resistance (NTSR). Based on our analysis of F2 seed families derived from two distinct blackgrass populations with the same NTSR phenotype, we demonstrate that the trait is polygenic and evolves from standing genetic variation. We present evidence that selection for NTSR has signatures of both parallel and non-parallel evolution. There are parallel and non-parallel changes at the transcriptional level of several stress- and defense-responsive gene families. At the genomic level, however, the genetic loci underpinning NTSR are different (non-parallel) between seed families. We speculate that variation in the number, regulation and function of stress- and defense-related gene families enable weedy species to rapidly evolve NTSR via exaptation of genes within large multi-functional gene families. These results provide novel insights into the potential for, and nature of plant adaptation in rapidly changing environments

RNA and protein biomarkers for detecting enhanced metabolic resistance to herbicides mesosulfuron-methyl and fenoxaprop-ethyl in black-grass (Alopecurus myosuroides)

BACKGROUND: The evolution of non-target site resistance (NTSR) to herbicides leads to a significant reduction in herbicide control of agricultural weed species. Detecting NTSR in weed populations prior to herbicide treatment would provide valuable information for effective weed control. While not all NTSR mechanisms have been fully identified, enhanced metabolic resistance (EMR) is one of the better studied, conferring tolerance through increased herbicide detoxification. Confirming EMR towards specific herbicides conventionally involves detecting metabolites of the active herbicide molecule in planta, but this approach is time consuming and requires access to well-equipped laboratories. RESULTS: In this study, we explore the potential of using molecular biomarkers to detect EMR before herbicide treatment in black-grass (Alopecurus myosuroides). We test the reliability of selected biomarkers to predict EMR, and survival after herbicide treatments in both reference and 27 field-derived black-grass populations collected from sites across the UK. The combined analysis of the constitutive expression of biomarkers, and metabolism studies confirmed three proteins namely, AmGSTF1, AmGSTU2 and AmOPR1, as differential biomarkers of EMR toward the herbicides fenoxaprop-ethyl and mesosulfuron in black-grass. CONCLUSION: Our findings demonstrate that there is potential to use molecular biomarkers to detect EMR toward specific herbicides in black-grass without reference to metabolism analysis. However, biomarker development must include testing at both transcript and protein levels in order to be reliable indicators of resistance. This work is a first step towards more robust resistance biomarker development, which could be expanded into other herbicide chemistries, for on-farm testing and monitoring EMR in uncharacterised black-grass populations

RNA and protein biomarkers for detecting enhanced metabolic resistance to herbicides mesosulfuron-methyl and fenoxaprop-ethyl in black-grass (<em>Alopecurus myosuroides</em>)

\ua9 2024 The Authors. Pest Management Science published by John Wiley &amp; Sons Ltd on behalf of Society of Chemical Industry. BACKGROUND: The evolution of non-target site resistance (NTSR) to herbicides leads to a significant reduction in herbicide control of agricultural weed species. Detecting NTSR in weed populations prior to herbicide treatment would provide valuable information for effective weed control. While not all NTSR mechanisms have been fully identified, enhanced metabolic resistance (EMR) is one of the better studied, conferring tolerance through increased herbicide detoxification. Confirming EMR towards specific herbicides conventionally involves detecting metabolites of the active herbicide molecule in planta, but this approach is time-consuming and requires access to well-equipped laboratories. RESULTS: In this study, we explored the potential of using molecular biomarkers to detect EMR before herbicide treatment in black-grass (Alopecurus myosuroides). We tested the reliability of selected biomarkers to predict EMR and survival after herbicide treatments in both reference and 27 field-derived black-grass populations collected from sites across the UK. The combined analysis of the constitutive expression of biomarkers and metabolism studies confirmed three proteins, namely, AmGSTF1, AmGSTU2 and AmOPR1, as differential biomarkers of EMR toward the herbicides fenoxaprop-ethyl and mesosulfuron in black-grass. CONCLUSION: Our findings demonstrate that there is potential to use molecular biomarkers to detect EMR toward specific herbicides in black-grass without reference to metabolism analysis. However, biomarker development must include testing at both transcript and protein levels in order to be reliable indicators of resistance. This work is a first step towards more robust resistance biomarker development, which could be expanded into other herbicide chemistries for on-farm testing and monitoring EMR in uncharacterised black-grass populations. \ua9 2024 The Authors. Pest Management Science published by John Wiley &amp; Sons Ltd on behalf of Society of Chemical Industry

Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin-vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrodinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrodinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.Comment: Published version with typos corrected. Significantly expanded version of a talk given by the first author at the 2008 BIRS workshop on "Geometric Flows in Mathematics and Physics