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On the hierarchies of the fully nonlinear Möbius-invariant and symmetry-integrable evolution equations of order three J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Marianna Euler; Norbert Euler
This is a follow-up paper to the results published in Studies in Applied Mathematics 143, 139–156 (2019), where we reported a classification of 3rd- and 5th-order semi-linear symmetry-integrable evolution equations that are invariant under the Möbius transformation, which includes a list of fully nonlinear 3rd-order equations that admit these properties. In the current paper we propose a simple method
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Minimal surfaces associated with orthogonal polynomials J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Vincent Chalifour; Alfred Michel Grundland
This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces approach defined by the Enneper-Weierstrass formula for immersion and the solutions of the Gauss-Weingarten equations for moving frames, we derive the three-dimensional
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Study on geometric structures on Lie algebroids with optimal control applications J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Esmaeil Peyghan; Liviu Popescu
We construct ρ£ -covariant derivatives in π*π as the generalization of covariant derivative in π*π to £π E. Moreover, we introduce Berwald and Yano derivatives as two important classes of ρ£ -covariant derivatives in π∗π and we study properties of them. Finally, we solve an optimal control problem using some geometric structures and Pontryagin Maximum Principle on Lie algebroids.
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Nonlocal symmetries and group invariant solutions for the coupled variable-coefficient Newell-Whitehead system J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Yarong Xia; Ruoxia Yao; Xiangpeng Xin
Starting from the Lax pairs, the nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are localized to Lie point symmetries and the coupled variable-coefficient Newell-Whitehead system is extended to an enlarged system with the auxiliary variable. Then the finite symmetry transformation
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Asymptotics behavior for the integrable nonlinear Schrödinger equation with quartic terms: Cauchy problem J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Lin Huang
We consider the Cauchy problem of integrable nonlinear Schrödinger equation with quartic terms on the line. The first part of the paper considers the Riemann-Hilbert formula via the unified method(also known as the Fokas method). The second part of the paper establishes asymptotic formulas for the solution of initial value problem using the nonlinear steepest descent method(also known as the Deift-Zhou
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On the discretization of Darboux Integrable Systems J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Kostyantyn Zheltukhin; Natalya Zheltukhina
We study the discretization of Darboux integrable systems. The discretization is done using x-, y-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
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Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Xiaoxue Xu; Cewen Cao; Guangyao Zhang
Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting to the dis- crete version of Liouville-Arnold theorem, finite genus solutions to the lSKdV equation
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Decomposition of 2-Soliton Solutions for the Good Boussinesq Equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Vesselin Vatchev
We consider decompositions of two-soliton solutions for the good Boussinesq equation obtained by the Hirota method and the Wronskian technique. The explicit forms of the components are used to study the dynamics of 2-soliton solutions. An interpretation in the context of eigenvalue problems arising from KdV type equations and transport equations is considered. Numerical examples are included.
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Integrability conditions of a weak saddle in generalized Liénard-like complex polynomial differential systems J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Jaume Giné; Claudia Valls
We consider the complex differential system ẋ = x + y f (x) , ẏ = −y + x f (y) , where f is the analytic function . This system has a weak saddle at the origin and is a generalization of complex Liénard systems. In this work we study its local analytic integrability.
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Symmetry classification of scalar Ito equations with multiplicative noise J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Giuseppe Gaeta; Francesco Spadaro
We provide a symmetry classification of scalar stochastic equations with multiplicative noise. These equations can be integrated by means of the Kozlov procedure, by passing to symmetry adapted variables.
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Gambier lattices and other linearisable systems J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Basil Grammaticos; Alfred Ramani
We propose two different appraoches to extending the Gambier mapping to a two-dimensional lattice equation. A first approach relies on a hypothesis of separate evolutions in each of the two directions. We show that known equations like the Startsev-Garifullin-Yamilov equation, the Hietarinta equation, as well as one proposed by the current authors, are in fact Gambier lattices. A second approach, based
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Trigonal Toda Lattice Equation J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-09-04 Shigeki Matsutani
In this article, we give the trigonal Toda lattice equation, for a lattice point as a directed 6-regular graph where , and its elliptic solution for the curve y(y–s) = x 3, (s ≠ 0).
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Global dynamics of a Lotka–Volterra system in ℝ3 J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 Jaume Llibre; Y. Paulina Martínez; Claudia Valls
In this work we consider the Lotka–Volterra system in ℝ3 introduced recently in [7], and studied also in [8] and [14]. In the first two papers the authors mainly studied the integrability of this differential system, while in the third paper they studied the system as a Hamilton- Poisson system, and also started the analysis of its dynamics. Here we provide the global phase portraits of this 3-dimensional
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Group analysis of the generalized Burnett equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 Alexander V. Bobylev; Sergey V. Meleshko
In this paper group properties of the so-called Generalized Burnett equations are studied. In contrast to the classical Burnett equations these equations are well-posed and therefore can be used in applications. We consider the one-dimensional version of the generalized Burnett equations for Maxwell molecules in both Eulerian and Lagrangian coordinates and perform the complete group analysis of these
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Nonautonomous symmetries of the KdV equation and step-like solutions J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 V.E. Adler
We study solutions of the KdV equation governed by a stationary equation for symmetries from the non-commutative subalgebra, namely, for a linear combination of the master-symmetry and the scaling symmetry. The constraint under study is equivalent to a sixth order nonautonomous ODE possessing two first integrals. Its generic solutions have a singularity on the line t = 0. The regularity condition selects
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Hierarchies of q-discrete Painlevé equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 Huda Alrashdi; Nalini Joshi; Dinh Thi Tran
In this paper, we construct a new hierarchy based on the third q-discrete Painlevé equation (qPIII) and also study the hierarchy of the second q-discrete Painlevé equation (qPII). Both hierarchies are derived by using reductions of the general lattice modified Korteweg-de Vries equation. Our results include Lax pairs for both hierarchies and these turn out to be higher degree expansions of the non-resonant
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Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 John Cobb; Alex Kasman; Albert Serna; Monique Sparkman
Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these three basic types. A complete characterization of the parameters that lead to non-singular 1soliton and periodic solutions is given. Surprisingly, it is shown that such
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On the global dynamics of a three-dimensional forced-damped differential system J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 Jaume Llibre; Y. Paulina Martínez; Claudia Valls
In this paper by using the Poincaré compactification of ℝ3 we make a global analysis of the model xʹ = ax + y + yz, yʹ = x – ay + bxz, zʹ = cz – bxy. In particular we give the complete description of its dynamics on the infinity sphere. For a + c = 0 or b = 1 this system has invariants. For these values of the parameters we provide the global phase portrait of the system in the Poincaré ball. We also
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Integrable Boundary Conditions for the Hirota-Miwa Equation and Lie Algebras J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 Ismagil Habibullin; Aigul Khakimova
Systems of discrete equations on a quadrilateral graph related to the series of the affine Lie algebras are studied. The systems are derived from the Hirota-Miwa equation by imposing boundary conditions compatible with the integrability property. The Lax pairs for the systems are presented. It is shown that in the continuum limit the quad systems tend to the corresponding systems of the differential
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Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-05-04 Zühal Küçükarslan Yüzbaşı; Stephen C. Anco
Elastic (stretching) flows of null curves are studied in three-dimensional Minkowski space. As a main tool, a natural type of moving frame for null curves is introduced, without use of the pseudo-arclength. This new frame is related to a Frenet null frame by a gauge transformation that belongs to the little group contained in the Lorentz group SO(2, 1) and provides an analog of the Hasimoto transformation
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Linearizable boundary value problems for the elliptic sine-Gordon and the elliptic Ernst equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Jonatan Lenells; Athanassios S. Fokas
By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic sine-Gordon equation, as well as for the elliptic version of the Ernst equation, can be expressed in terms of the solution of appropriate 2×2-matrix Riemann–Hilbert
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Induced Dynamics J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 A. K. Pogrebkov
Construction of new integrable systems and methods of their investigation is one of the main directions of development of the modern mathematical physics. Here we present an approach based on the study of behavior of roots of functions of canonical variables with respect to a parameter of simultaneous shift of space variables. Dynamics of singularities of the KdV and Sinh–Gordon equations, as well
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Solutions of the constrained mKP hierarchy by Boson-Fermion correspondence J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Huizhan Chen; Lumin Geng; Jipeng Cheng
In this paper, the Hirota bilinear equation of the constrained modified KP hierarchy is expressed as the vacuum expectation values of Clifford operators by using the free fermions method of mKP hierarchy. Then we mainly use the Boson-Fermion correspondence to solve the Hirota bilinear equation of the k-constrained mKP hierarchy. Further, by choosing special group elements in GL∞, the corresponding
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Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Isaac A. García; Benito Hernández-Bermejo
A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the bifurcation phenomena of periodic orbits as a result of these perturbations in the period annulus associated to the unperturbed harmonic oscillator. This is accomplished
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A generalization of the Landau-Lifschitz equation: breathers and rogue waves J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Ruomeng Li; Xianguo Geng; Bo Xue
A generalization of the Landau-Lifschitz equation with uniaxial anisotropy is proposed, which can also reduce to the derivative nonlinear Schrödinger equation under an infinitesimal parameter. Based on the gauge transformation between Lax pairs, an N-fold generalized Darboux transformation is constructed for the generalization of the Landau-Lifschitz equation with uniaxial anisotropy. As applications
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Final evolutions of a class of May-Leonard Lotka-Volterra systems J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Claudio A. Buzzi; Robson A. T. Santos; Jaume Llibre
We study a particular class of Lotka-Volterra 3-dimensional systems called May-Leonard systems, which depend on two real parameters a and b, when a + b = −1. For these values of the parameters we shall describe its global dynamics in the compactification of the non-negative octant of ℝ3 including its infinity. This can be done because this differential system possesses a Darboux invariant.
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Canonical spectral coordinates for the Calogero-Moser space associated with the cyclic quiver J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Tamás Görbe; Ádám Gyenge
Sklyanin’s formula provides a set of canonical spectral coordinates on the standard Calogero-Moser space associated with the quiver consisting of a vertex and a loop. We generalize this result to Calogero-Moser spaces attached to cyclic quivers by constructing rational functions that relate spectral coordinates to conjugate variables. These canonical coordinates turn out to be well-defined on the corresponding
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Symmetry reductions and new functional separable solutions of nonlinear Klein–Gordon and telegraph type equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Alexei I. Zhurov; Andrei D. Polyanin
The paper is concerned with different classes of nonlinear Klein–Gordon and telegraph type equations with variable coefficients c(x)utt + d(x)ut = [a(x)ux]x + b(x)ux + p(x) f (u), where f (u) is an arbitrary function. We seek exact solutions to these equations by the direct method of symmetry reductions using the composition of functions u = U (z) with z = φ (x, t). We show that f (u) and any four
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Cusped solitary wave with algebraic decay governed by the equation for surface waves of moderate amplitude J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Bo Jiang; Youming Zhou
The existence of a new type of cusped solitary wave, which decays algebraically at infinity, for a nonlinear equation modeling the free surface evolution of moderate amplitude waves in shallow water is established by employing qualitative analysis for differential equations. Furthermore, the exact parametric representation as well as its planar graph for such type of wave is also given.
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On constant solutions of SU(2) Yang-Mills equations with arbitrary current in Euclidean space ℝn J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Dmitry Shirokov
In this paper, we present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space ℝn of arbitrary finite dimension n. Using the invariance of the Yang-Mills equations under the orthogonal transformations of coordinates and gauge invariance, we choose a specific system of coordinates and a specific gauge fixing for
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The Lie symmetry group of the general Liénard-type equation J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2020-01-27 Ágota Figula; Gábor Horváth; Tamás Milkovszki; Zoltán Muzsnay
We consider the general Liénard-type equation . This equation naturally admits the Lie symmetry . We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions f0, . . . , fn. Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases n = 1 or n
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Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 Francesco Calogero, Farrin Payandeh
In this paper we identify certain peculiar systems of 2 discrete-time evolution equations, which are algebraically solvable. Here l is the “discrete-time” independent variable taking integer values (l = 0, 1, 2, . . .), xn ≡ xn(l) are 2 dependent variables, and are the corresponding 2 updated variables. In a previous paper the 2 functions F(n)(x1, x2), n = 1, 2, were defined as follows: F(n)(x1, x2)
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Constructing discrete Painlevé equations: from E8(1) to A1(1) and back J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 A. Ramani, B. Grammaticos, R. Willox, T. Tamizhmani
The ‘restoration method’ is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that are equivalent to it by homographic transformations. Discrete Painlevé equations are then obtained by deautonomising
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On Slant Magnetic Curves in S-manifolds J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 Şaban Güvenç, Cihan Özgür
We consider slant normal magnetic curves in (2n + 1)-dimensional S-manifolds. We prove that γ is a slant normal magnetic curve in an S-manifold (M2m+s, φ, ξα, ηα, g) if and only if it belongs to a list of slant φ-curves satisfying some special curvature equations. This list consists of some specific geodesics, slant circles, Legendre and slant helices of order 3. We construct slant normal magnetic
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A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 Alexander G. Rasin, Jeremy Schiff
We study the simple-looking scalar integrable equation fxxt 3( fx ft 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries
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On the global dynamics of the Newell–Whitehead system J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 Claudia Valls
In this paper by using the Poincaré compactification in ℝ3 we make a global analysis of the model x′ = z, y′ = b(x – dy), z′ = x(x2 – 1)+ y + cz with b ϵ ℝ and c, d ϵ ℝ+, here known as the three-dimensional Newell–Whitehead system. We give the complete description of its dynamics on the sphere at infinity. For some values of the parameters this system has invariant algebraic surfaces and for these
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An exact solution for geophysical internal waves with underlying current in modified equatorial β -plane approximation* * The work is supported in part by a NSFC Grant No. 11531006, PAPD of Jiangsu Higher Education Institutions, and the Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Applications. J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 Dong Su, Hongjun Gao
In this paper, a modification of the standard geophysical equatorial β-plane model equations, incorporating a gravitational-correction term in the tangent plane approximation, is derived. We present an exact solution to meet the modified governing equations, whose form is explicit in the Lagrangian framework and which represents internal oceanic waves in the presence of a constant underlying current
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Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 M.E. Fels, E. Yaşar
For a scalar evolution equation ut = K(t, x, u, ux, . . . , u2m+1) with m ≥ 1, the cohomology space H1,2() is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for ut = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H1,2() and subsequently can be naturally identified with the space
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Systems of Hamilton-Jacobi equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-07-09 Julio Cambronero, Javier Pérez Álvarez
In this article we develop a generalization of the Hamilton-Jacobi theory, by considering in the cotangent bundle an involutive system of dynamical equations.
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A Note on the Equivalence of Methods to finding Nonclassical Determining Equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 J. Goard
In this note we prove that the method of Bîlã and Niesen to determine nonclassical determining equations is equivalent to that of Nucci’s method with heir-equations and thus in general is equivalent to using an appropriate form of generalised conditional symmetry.
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Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Rustem N. Garifullin, Giorgio Gubbiotti, Ravil I. Yamilov
In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among
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Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Jarmo Hietarinta
We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of “multidimensional consistency” (MDC): it should be possible to extend the equation from two to higher dimensions so that the embedded two-dimensional lattice
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Two-component generalizations of the Novikov equation J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Hongmin Li
Some two-component generalizations of the Novikov equation, except the Geng-Xue equation, are presented, as well as their Lax pairs and bi-Hamiltonian structures. Furthermore, we study the Hamiltonians of the Geng-Xue equation and discuss the homogeneous and local properties of them.
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Bilinear Identities and Squared Eigenfunction Symmetries of the BCr-KP Hierarchy J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Lumin Geng, Huizhan Chen, Na Li, Jipeng Cheng
The BCr-KP hierarchy is an important sub hierarchy of the KP hierarchy, which includes the BKP and CKP hierarchies as the special cases. Some properties of the BCr-KP hierarchy and its constrained case are investigated in this paper, including bilinear identities and squared eigenfunction symmetries. We firstly discuss the bilinear identities of the BCr-KP hierarchy, and then generalize them into the
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Products in the category of -manifolds J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Andrew Bruce, Norbert Poncin
We prove that the category of -manifolds has all finite products. Further, we show that a -manifold (resp., a -morphism) can be reconstructed from its algebra of global -functions (resp., from its algebra morphism between global -function algebras). These results are of importance in the study of Lie groups. The investigation is all the more challenging, since the completed tensor product of the structure
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Integration of the stochastic logistic equation via symmetry analysis J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Giuseppe Gaeta
We apply the recently developed theory of symmetry of stochastic differential equations to a stochastic version of the logistic equation, obtaining an explicit integration, i.e. an explicit formula for the process in terms of any single realization of the driving Wiener process.
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Quasiperiodic Solutions of the Heisenberg Ferromagnet Hierarchy J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Peng Zhao, Engui Fan, Temuerchaolu
We present quasi-periodic solutions in terms of Riemann theta functions of the Heisenberg ferromagnet hierarchy by using algebro-geometric method. Our main tools include algebraic curve and Riemann surface, polynomial recursive formulation and a special meromorphic function.
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The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-05-09 Boling Guo, Nan Liu
We derive the long-time asymptotics for the solution of initial-boundary value problem of coupled nonlinear Schrödinger equation whose Lax pair involves 3 × 3 matrix in present paper. Based on a nonlinear steepest descent analysis of an associated 3 × 3 matrix Riemann–Hilbert problem, we can give the precise asymptotic formulas for the solution of the coupled nonlinear Schrödinger equation on the half-line
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Liouvillian integrability of a general Rayleigh-Duffing oscillator J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Jaume Giné, Claudia Valls
We give a complete description of the Darboux and Liouville integrability of a general Rayleigh-Duffing oscillator through the characterization of its polynomial first integrals, Darboux polynomials and exponential factors.
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Generalized Solvable Structures and First Integrals for ODEs Admitting an Symmetry Algebra J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Paola Morando, Concepción Muriel, Adrián Ruiz
The notion of solvable structure is generalized in order to exploit the presence of an algebra of symmetries for a kth-order ordinary differential equation with k > 3. In this setting, the knowledge of a generalized solvable structure for allows us to reduce to a family of second-order linear ordinary differential equations depending on k − 3 parameters. Examples of explicit integration of fourth and
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A construction of Multidimensional Dubrovin-Novikov Brackets J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Ian A. B. Strachan
A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined from com- mutative, associative algebras. Given such an algebra, the construction involves only linear
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Delta shock waves in conservation laws with impulsive moving source: some results obtained by multiplying distributions J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 C.O.R. Sarrico
The present paper concerns the study of a Riemann problem for the conservation law ut + [ϕ (u)]x = kδ (x − vt) where x, t, k, v and u = u(x,t) are real numbers. We consider ϕ an entire function taking real values on the real axis and δ stands for the Dirac measure. Within a convenient space of distributions we will explicitly see the possible emergence of waves with the shape of shock waves, delta
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A Hamiltonian yielding damped motion in an homogeneous magnetic field: quantum treatment J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 François Leyvraz, Francesco Calogero
In earlier work, a Hamiltonian describing the classical motion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated. Here we derive the quantum propagator for the Hamiltonian in different representations, one corresponding to momentum space, the other to position, and the third
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Bilinear identities for the constrained modified KP hierarchy J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Huizhan Chen, Lumin Geng, Na Li, Jipeng Cheng
In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ.
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Analytical Cartesian solutions of the multi-component Camassa-Holm equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Hongli An, Liying Hou, Manwai Yuen
Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t)+A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing
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Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Francesco Calogero, Farrin Payandeh
The evolution equations mentioned in the title of this paper read as follows: where ℓ is the “discrete-time” independent variable taking integer values (ℓ = 0, 1, 2, … ), xn ≡ xn(ℓ) are the 2 dependent variables, , and the 2 functions P(n)(x1, x2), n = 1, 2, are 2 polynomials in the 2 dependent variables x1(ℓ) and x2(ℓ). The results reported in this paper have been obtained by an appropriate modification
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Differential Equations Invariant Under Conditional Symmetries J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Decio Levi, Miguel A. Rodríguez, Zora Thomova
Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous Korteweg–de Vries like equations are given to show and clarify the methodology introduced.
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N = 2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Ziemowit Popowicz
The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures
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Moving Boundary Problems for Heterogeneous Media. Integrability via Conjugation of Reciprocal and Integral Transformations J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2019-03-11 Colin Rogers
The combined action of reciprocal and integral-type transformations is here used to sequentially reduce to analytically tractable form a class of nonlinear moving boundary problems involving heterogeneity. Particular such Stefan problems arise in the description of the percolation of liquids through porous media in soil mechanics.
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A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation J. Nonlinear Math. Phys. (IF 0.978) Pub Date : 2018-12-03 Xianguo Geng, Yunyun Zhai, Bo Xue, Jiao Wei
Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties