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In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we study a family of integrable equations: the nonabelian Narita–Itoh–Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz–Ladik lattice. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of pulse solutions in a system with interacting instability mechanisms, which is described by a Ginzburg-Landau equation for an A-mode, coupled to a diffusion equation for a B-mode. Our main question is whether this coupling may stabilize solutions of the Ginzburg-Landau equation that are unstable when the interactions with the neutrally stable B-mode are not included in the model. The spatially homogeneous B-mode is supposed to be neutrally stable. This implies that the pulse solutions cannot decay exponentially, but must decay with an algebraic rate as x? ±?. As a consequence, the methods that exist in the literature by which the stability of pulses in singularly perturbed reaction-diffusion systems can be studied need to be extended. This results in an 'algebraic NLEP approach', which is expected to be relevant beyond the setting of this paper. As in the case of a (weakly) stable B-mode (Doelman et al2004 J. Nonlin. Sci.14237-78) we establish by the application of this approach that the B-mode indeed introduces a mechanism that may stabilize pulses that are unstable when the interactions with the B-mode are not taken into account. [ABSTRACT FROM AUTHOR]

In this paper we consider the slightly L2-supercritical gKdV equations , with the nonlinearity and . In the previous work of the author, we know that there exists a stable self-similar blow-up dynamics for slightly L2-supercritical gKdV equations. Such solutions can be viewed as solutions with a single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution of the vortex filament equation, with X(s, 0) being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that X(s, t) is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gauß sum.We also study the fractal behaviour of X(0, t), relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal. [ABSTRACT FROM AUTHOR]

This paper studies the entropy of tree-shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree-shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of the binary Markov tree-shifts over two symbols is either 0 or . Meanwhile, the realization of a class of reals including multinacci numbers is elaborated, which indicates that tree-shifts are capable of rich phenomena. By considering the influence of three different types of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient conditions for the coincidence of entropy with and without boundary conditions are addressed. [ABSTRACT FROM AUTHOR]

In this paper we produce seven new algebras as confluences of the Cherednik algebra of type and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory. [ABSTRACT FROM AUTHOR]

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. Recurrences with this property appear in diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmüller theory and dimer models. In many cases where such recurrences appear, there is a common structural thread running between these different areas, in the form of Fomin and Zelevinsky's theory of cluster algebras. Laurent phenomenon algebras, as defined by Lam and Pylyavskyy, are an extension of cluster algebras, and share with them the feature that all the generators of the algebra are Laurent polynomials in any initial set of generators (seed). Here we consider a family of nonlinear recurrences with the Laurent property, referred to as 'Little Pi', which was derived by Alman et al via a construction of periodic seeds in Laurent phenomenon algebras, and generalizes the Heideman–Hogan family of recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. We derive the latter from linear relations with periodic coefficients, which were found recently by Kamiya et al from travelling wave reductions of a linearizable lattice equation on a six-point stencil. By making use of the periodic coefficients, we further show that the birational maps corresponding to the Little Pi family are maximally superintegrable. We also introduce another linearizable lattice equation on the same six-point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the six-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property. [ABSTRACT FROM AUTHOR]

In this paper, we study propagation in a non-local reaction–diffusion–mutation model describing the invasion of cane toads in Australia (Phillips et al 2006 Nature439 803). The population of toads is structured by a space variable and a phenotypical trait and the space diffusivity depends on the trait. We use a Schauder topological degree argument for the construction of some travelling wave solutions of the model. The speed c* of the wave is obtained after solving a suitable spectral problem in the trait variable. An eigenvector arising from this eigenvalue problem gives the flavour of the profile at the edge of the front. The major difficulty is to obtain uniform L∞ bounds despite the combination of non-local terms and a heterogeneous diffusivity. [ABSTRACT FROM AUTHOR]

We study a 4th order degenerate parabolic PDE model in one-dimension with a 2nd order correction modeling the evolution of a crystal surface under the influence of both thermal fluctuations and evaporation/deposition effects. First, we provide a non-rigorous derivation of the PDE from an atomistic model using variations on kinetic Monte Carlo rates proposed by the last author with Weare [Marzuola J L and Weare J 2013 Phys. Rev. E 88 032403]. Then, we prove the existence of a global in time weak solution for the PDE by regularizing the equation in a way that allows us to apply the tools of Bernis–Friedman [Bernis F and Friedman A 1990 J. Differ. Equ. 83 179–206]. The methods developed here can be applied to a large number of 4th order degenerate PDE models. In an appendix, we also discuss the global smooth solution with small data in the Weiner algebra framework following recent developments using tools of the second author with Robert Strain [Liu J G and Strain R M 2019 Interfaces Free Boundaries 21 51–86]. [ABSTRACT FROM AUTHOR]

For the Liénard systems \dot x=y,\,\,\dot y=-f_m(x)y-g_n(x) with fm and gn polynomials of degree m and n, respectively, we present explicit systems having algebraic limit cycles in the cases m [?] 2 and n [?] 2m + 1 and m [?] 3 and n = 2m. Also we prove that the Liénard system for m = 3 and n = 5 has no hyperelliptic limit cycles. This shows that the result of theorem 1(c) of Zoladek (1998 Trans. Am. Math. Soc. 350 1681-701) on the existence of algebraic limit cycles of the Liénard system is not correct. Moreover, we characterize all hyperelliptic limit cycles of the Liénard systems for m = 4 and n = 6 or n = 7. For m > 4 and n = 2m [?] 1 or n = 2m [?] 2 we prove that there are Liénard systems which have [m/2] [?] 1 algebraic limit cycles, where the [·] denotes the integer part function. [ABSTRACT FROM AUTHOR]

We study the deformations of two-component non-semisimple Poisson pencils of hydrodynamic type associated with Balinskiǐ–Novikov algebras. We show that in most cases the second order deformations are parametrized by two functions of a single variable. We find that one function is invariant with respect to the subgroup of Miura transformations, preserving the dispersionless limit, and another function is related to a one-parameter family of truncated structures. In two exceptional cases the second order deformations are parametrized by four functions. Among these two are invariants and two are related to a two-parameter family of truncated structures. We also study the lift of the deformations of n-component semisimple structures. This example suggests that deformations of non-semisimple pencils corresponding to the lifted invariant parameters are unobstructed. [ABSTRACT FROM AUTHOR]

A special case of coupled integrable nonlinear equations with a singular dispersion law is derived in the context of the small amplitude limit of general wave equations in a fluid-type warm electrons/cold ions plasma irradiated by a continuous laser beam. This model accounts for a nonlinear mode coupling of the electrostatic wave with the ion sound wave and is shown to be highly unstable. Its instability is understood as a continuous secular transfer of energy from the electrostatic wave to the ion sound wave through the ponderomotive force. The exact asymptotic solution of the system is constructed and shows that the dynamics of the energy transfer results in a singular asymptotic behavior of the ion sound wave, which explains the low penetration of the incident laser beam. [ABSTRACT FROM AUTHOR]

We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian–Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order—this implies Darboux integrability. On the other hand, the recurrent formulae for the leading principal minors are equivalent to the 2D Toda equation and its semi-discrete and lattice analogues with particular boundary conditions (cut-off constraints). This connection is used to obtain recurrent formulae and closed-form expressions for integrals of the Toda-type equations from the integrals of the determinantal equations. General solutions of the equations corresponding to vanishing determinants are given explicitly while, in the non-vanishing case, they are given in terms of solutions of ordinary linear equations. [ABSTRACT FROM AUTHOR]

We consider the discretely self-similar blowup solutions of the three-dimensional Navier–Stokes equations, and under suitable assumptions we show some estimates on the asymptotic behavior of the possible nontrivial velocity profiles. [ABSTRACT FROM AUTHOR]

We study the analytic properties of a matrix discrete system introduced by Cassatella and Mañas (2012 Stud. Appl. Math.128 252–74). The singularity confinement for this system is shown to hold generically, i.e. in the whole space of parameters except possibly for algebraic subvarieties. This paves the way to a generalization of Painlevé analysis to discrete matrix models. [ABSTRACT FROM AUTHOR]

We prove the existence of transition fronts for a large class of reaction–diffusion equations in one dimension, with inhomogeneous monostable reactions. We construct these as perturbations of corresponding front-like solutions to the linearization of the PDE at u = 0. While a close relationship between the solutions to the two PDEs is well known and has been exploited for KPP reactions (and our method is an extension of such ideas from Zlatoš A 2012 (J. Math. Pure Appl.98 89–102)), to the best of our knowledge this is the first time such an approach has been used in the construction and study of fronts for non-KPP monostable reactions. [ABSTRACT FROM AUTHOR]