Dispersionless Integrable Systems Research Articles

Higher-order reductions of the Mikhalev system

We consider the 3D Mikhalev system, ut=wx,uy=wt-uwx+wux,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ u_t=w_x, \\quad u_y= w_t-u w_x+w u_x, $$\\end{document}which has first appeared in the context of KdV-type hierarchies. Under the reduction w=f(u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w=f(u)$$\\end{document}, one obtains a pair of commuting first-order equations, ut=f′ux,uy=(f′2-uf′+f)ux,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ u_t=f'u_x, \\quad u_y=(f'^2-uf'+f)u_x, $$\\end{document}which govern simple wave solutions of the Mikhalev system. In this paper we study higher-order reductions of the form w=f(u)+ϵa(u)ux+ϵ2[b1(u)uxx+b2(u)ux2]+⋯,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ w=f(u)+\\epsilon a(u)u_x+\\epsilon ^2[b_1(u)u_{xx}+b_2(u)u_x^2]+\\cdots , $$\\end{document}which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at ϵn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon ^n$$\\end{document} are assumed to be differential polynomials of degree n in the x-derivatives of u. We will view w as an (infinite) formal series in the deformation parameter ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}. It turns out that for such a reduction to be non-trivial, the function f(u) must be quadratic, f(u)=λu2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(u)=\\lambda u^2$$\\end{document}, furthermore, the value of the parameter λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, λ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda =1$$\\end{document} and λ=3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda =3/2$$\\end{document}, as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of linear degeneracy of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations.

Superfield Bäcklund and Darboux transformations of an $\mathcal N=1$ supersymmetric coupled dispersionless integrable system

В терминах суперполевой матрицы Дарбу исследовано преобразование Дарбу $\mathcal N=1$ суперсимметричной связанной бездисперсионной интегрируемой системы. Для получения суперполевых $N$-солитонных решений этой системы использовано понятие квазидетерминантов. С использованием суперполевого представления Лакса получены суперполевые уравнения Риккати, из которых выводится суперполевое преобразование Беклунда для исследуемой системы. Преобразование Беклунда и преобразование Дарбу затем используются для нахождения явных выражений для суперполевых солитонных решений суперсимметричной связанной бездисперсионной интегрируемой системы.

Symmetry study of a novel integrable supersymmetric dispersionless system

A novel integrable supersymmetric dispersionless fermion system is proposed and studied by means of symmetry approach and the bosonization method, which is integrable under the meaning of possessing infinitely many higher order symmetries. It is shown that there are some types of infinitely many symmetries with arbitrary functions. The first type of infinitely many symmetries commutes each other, whereas the second type of higher order symmetries with arbitrary functions are not commutable and the commutators yield new higher order symmetries. By using the bosonization approach, the system can be bosonized to some special dark integrable systems which can be exactly solved in general.

CTE Solvability, Nonlocal Symmetry, and Interaction Solutions of Coupled Integrable Dispersionless System

The consistent tanh expansion (CTE) method is successfully applied to the coupled integrable dispersionless (CID) system. A nonauto-Bäcklund transformation (BT) theorem includes two fields f and v 1 is obtained by using the CTE method. One obtains the consistent condition in the nonauto-BT theorem by means of the relation between the fields f and v 1 . The CID system possesses the CTE solvability property by some detailed analysis. Many interactions between one soliton and multiple resonant solitons, and between one soliton and cnoidal waves are generated by using the nonauto-BT theorem. The types of bright and gray two front waves are shown by some figures. In the meanwhile, the nonlocal symmetry is obtained by the truncated Painlevé method and the Möbious invariant form. The initial value problem and an auto-BT are constructed by the localization procedure.

The two-component complex coupled integrable dispersionless equations: Darboux transformation and soliton solutions

The complex coupled integrable dispersionless (CID)system is an important system. It has been shown that complex CID equation is gauge equivalent to the Pohlmeyer–Lund–Regge model (a complex sine-Gordon equation) and can also be transformed to the complex short pulse equation. CID system can describe a current-fed string interacting with an external magnetic field. In this paper, we investigate the two-component complex coupled integrable dispersionless system. By using the Darboux transformation, we obtain a new type of solution (the breather-like soliton solution) with zero boundary condition and a large number of solutions (the breather-II type solution, bright–dark soliton solution, the resonance solutions and the semi-rational solutions) with non-zero boundary condition. In particular, these solutions are not available for the scalar complex integrable dispersionless system.

On a generalization of a coupled integrable dispersionless system

In this paper, we are concerned with a generalization of a coupled integrable dispersionless system, which plays a crucial role in nonlinear problems. We show with the appropriate transformations that the solutions can be expressed in terms of those of the sine-Gordon and sinh-Gordon equations involving an arbitrary function. The exact solutions are investigated.

Matrix extension of multidimensional dispersionless integrable hierarchies

We consistently develop a recently proposed scheme of matrix extension of dispersionless integrable systems for the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax pairs, Lax-Sato equations, matrix equations on the background of vector fields and the dressing scheme. Reductions, construction of solutions and connections to geometry are discussed. We consider separately a case of Abelian extension, for which the Riemann-Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of Penrose formula in the curved space.

SOLITARY WAVE SOLUTIONS FOR SPACE-TIME FRACTIONAL COUPLED INTEGRABLE DISPERSIONLESS SYSTEM VIA GENERALIZED KUDRYASHOV METHOD

In this article, space-time fractional coupled integrable dispersionless system is considered, and we use fractional derivative in the sense of modified Riemann-Liouville. The fractional system has reduced to an ordinary differential system by fractional transformation and the generalized Kudryashov method is applied to obtain exact solutions. We also testify performance as well as precision of the applied method by means of numerical tests for obtaining solutions. The obtained results have been graphically presented to show the properties of the solutions.

Soliton solutions of the semi-discrete complex coupled dispersionless integrable system

In this paper, a semi-discrete complex coupled dispersionless integrable system is proposed. The bright soliton solution, breather solution and rogue wave solution of the semi-discrete complex coupled dispersionless integrable system are obtained by the Darboux transformation method. The collision behavior between bright solitary waves is studied through asymptotic analysis.

Dispersionless integrable systems and the Bogomolny equations on an Einstein–Weyl geometry background

We derive a dispersionless integrable system describing a local form of a general three-dimensional Einstein-Weyl geometry with an Euclidean (positive) signature, construct its matrix extension and demonstrate that it leads to the Bogomolny equations for a non-abelian monopole on an Einstein-Weyl geometry background. The corresponding dispersionless integrable hierarchy, its matrix extension and the dressing scheme are also considered.

Novel solutions of general and reverse space-time nonlocal coupled integrable dispersionless systems

In this paper, we have studied a general coupled integrable dispersionless (CID) system and obtained nontrivial solutions in terms of the ratio of determinants by using matrix Darboux transformation. We obtain symmetry non-preserving oscillating solutions of the general CID system in zero background. By using suitable reduction conditions we obtain symmetry preserving unstable soliton solutions for reverse space-time nonlocal CID system. We also recovered stable soliton solutions for the classical CID system under local symmetry reduction condition. Furthermore, the dynamics of symmetry breaking and symmetry preserving first- and second-order nontrivial solutions for generalized, reverse space-time nonlocal and classical CID systems have been presented.

On $$\varvec{\mathcal {PT}}$$-symmetric semi-discrete coupled integrable dispersionless system

In this work, we have proposed a $$\mathcal {PT}$$-symmetric reverse space-time nonlocal semi-discrete coupled integrable dispersionless system by discretization of associated linear eigenvalue problem. Darboux transformation has been applied to construct nontrivial solutions in terms of determinants. To elaborate the dynamics of solutions, explicit expressions of first two nontrivial solutions are computed. We analyze both symmetry broken and preserving solutions for generalized and $$\mathcal {PT}$$-symmetric reverse space-time nonlocal semi-discrete coupled integrable dispersionless systems, respectively. We also obtain first two nontrivial soliton solutions of local semi-discrete coupled integrable dispersionless system under certain condition on the spectral parameters.

Bilinearization and Soliton Solutions of a Supersymmetric Multicomponent Coupled Dispersionless Integrable System

We present a supersymmetric generalization of a multicomponent coupled dispersionless (SUSY-MCD) integrable system. Expanding the superfields of the SUSY-MCD system, we obtain the component fermionic and bosonic held equations. We propose a bilinear form of the SUSY-MCD system and find soliton solutions of it.

Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

Dispersionless integrable hierarchies and GL(2, ℝ) geometry

AbstractParaconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

The dispersionless completely integrable heavenly type Hamiltonian flows and their differential-geometric structure

There are reviewed modern investigations devoted to studying nonlinear dispersiveless heavenly type integrable evolutions systems on functional spaces within the modern differential-geometric and algebraic tools. Main accent is done on the loop diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, generating dispersionless heavenly type integrable systems.

Integrability of a Multicomponent Coupled Dispersionless Integrable System

We present a multicomponent coupled dispersionless integrable system and show that it is integrable in the sense of the existence of a Lax pair representation and also the existence of an infinite sequence of conserved quantities, a Darboux transformation, and soliton solutions.

Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application

Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were initiated by Plebanski and later analyzed in a series of articles. The AKS-algebraic and related $\mathcal{R}$-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie--Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equations under consideration. Moreover, all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a fascinating Lagrange--d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax--Sato equations is also discussed. We pay a special attention to a generalization of the devised Lie-algebraic scheme to a case of loop Lie superalgebras of superconformal diffeomorphisms of the $1|N$-dimensional supertorus. This scheme is applied to constructing the Lax--Sato integrable supersymmetric analogs of the Liouville and Mikhalev-Pavlov heavenly equation for every $N\in\mathbb{N}\backslash\lbrace 4;5\rbrace.$

Quasideterminant solutions of 2-component non–commutative complex coupled integrable dispersionless system

We present a non–commutative extension of two-component complex coupled integrable disspersionless (NC-CCID) system. We studied matrix Darboux transformation and generate multiple soliton solutions by using the notation of quasideterminants. Further, we obtain single and double soliton solutions in terms of quasideterminants for NC-CCID system. In the commutative limit, we obtain soliton solutions for CCID system as ratio of ordinary determinants.

Integrability Properties of A Supersymmetric Coupled Dispersionless Integrable System

We study the integrability aspects of an N=1 supersymmetric coupled dispersionless (SUSY-CD) integrable system in detail. We present a superfield Lax representation of the SUSY-CD system by writing its (3×3)-matrix superfield Lax pair and show that the zero-curvature condition corresponds to the SUSY-CD system. From the fermionic superfield Lax representation, we obtain a set of coupled superfield Riccati equations that we further use to obtain an infinite set of superfield conserved currents. We investigate the Darboux transformation of the SUSY-CD system and use it to obtain multisoliton solutions of the system.