We present a systematic way to construct solutions of the (n=5)-reduction of the BKP and CKP hierarchies from the general tau function of the KP hierarchy. We obtain the one-soliton, two-soliton, and periodic solution for the bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK) and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions such as left- and right-going solitons are classified according to the symmetries of the 5th roots of exp(i epsilon). Furthermore, we show that the soliton solutions of the n-reduction of the BKP and CKP hierarchies with n= 2 j +1, j=1, 2, 3, ..., can propagate along j directions in the 1+1 space-time domain. Each such direction corresponds to one symmetric distribution of the nth roots of exp(i epsilon). Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian tau function of the sub-hierarchies BKP and CKP. If n is even, we again find two-peak solitons. Last, we obtain the "stationary" soliton for the higher-order KP hierarchy.
It is shown that a set of functions which characterise the Lax hierarchy of non-linear equations may be represented in terms of the eigenstates of the potential which satisfies the generalised KdV equation. Such a representation leads to sumrules relating integrals involving the soliton potential and its various derivatives to sums involving the boundstate eigenvalues of the Schroedinger equation for the reflectionless potential. A new hierarchy of functions, which is in a sense dual to the Lax hierarchy, is identified. It is shown that time dependent equations involving the dual functions may…
A counterpart of the Wadati-Konno-Ichikawa (WKI) soliton hierarchy, associated with so(3,R), is presented through the zero curvature formulation. Its spectral matrix is defined by the same linear combination of basis vectors as the WKI one, and its Hamiltonian structures yielding Liouville integrability are furnished by the trace identity.
It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator.We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the $n$-soliton solutions of the KdV hierarchy can be completely depicted by a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. T…
In the recent paper (Wen-Xiu Ma, Solomon Manukure and Hong-Chan Zheng, arXiv:1405.1089), the authors proposed an integrable hierarchy different from the well-known Wadati-Konno-Ichikawa (WKI) hierarchy. However, using a simple linear change of dependent variables, one can check that their hierarchy is equivalent to the WKI hierarchy. For the same reason, some "new integrable" hierarchies proposed by Wen-Xiu Ma and coworkers in recent e-prints are equivalent to the already known ones.
We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one…
We investigate soliton solutions of the Toda hierarchy on a quasi-periodic finite-gap background by means of the double commutation method and the inverse scattering transform. In particular, we compute the phase shift caused by a soliton on a quasi-periodic finite-gap background. Furthermore, we consider short range perturbations via scattering theory. We give a full description of the effect of the double commutation method on the scattering data and establish the inverse scattering transform in this setting.
It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the nu…
We classify the soliton data in the totally non--negative part of Gr(k,n) which may be associated to algebraic-geometric data on certain rational degenerations of regular hyperelliptic M-curves. Such degenerate rational curve G is a desingularization of that constructed in Abenda-Grinevich (arXiv:1506.00563) for soliton data in the totally positive part of Gr(n-1,n) and the KP wavefunctions are the same in such case. G is also the curve constructed in the paper for soliton data in the totally positive part of Gr(1,n). For any such G and for any fixed k between 2 and (n-2), we show that k-compa…
In the present paper, we are concerned with the tau function and its connection with the Kadomtsev-Petviashvili (KP) theory for the massive Thirring (MT) model. First, we bilinearize the massive Thirring model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP-Toda hierarchy, we derive the multi-bright solution to the MT model by the KP hierarchy reductions. Then, we show that the discrete KP equation can generate a set of bilinear equations of a deformed KP-Toda hierarchy through Miwa transformation. By imposing constra…
A Kac-Moody algebra construction for the integrable hierarchy containing the Gardner equation is proposed. Solutions are systematically constructed employing the dressing method and deformed vertex operators which takes into account the nonvanishing boundary value problem for the mKdV hierarchy. Explicit examples are given and besides usual KdV like solitons, our solutions contemplate the large amplitude table-top solitons, kinks, dark solitons, breathers and wobbles.
Using the determinant representation of gauge transformation operator, we have shown that the general form of $τ$ function of the $q$-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP ($q$-cKP) hierarchy, i.e. $l$-constraints of $q$-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of $q$-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear $q$-partial differential equati…
We aim to present and analyze a nonlinear nonlocal reverse-spacetime fifth-order scalar Sasa-Satsuma equation, based on a nonlocal $5 \times 5$ matrix AKNS spectral problem. Starting from a nonlocal matrix AKNS spectral problem, local and nonlocal symmetry relations are derived from a group of rotations. A kind of Riemann-Hilbert problem is formulated, which allows to generate soliton solutions by using vectors lying in the kernel of the matrix Jost solutions. When reflection coefficients are zeros, the jump matrix is the identity and the corresponding Riemann-Hilbert problem yields soliton so…
Employing the Hirota's method, a class of soliton solutions for the N=2 super mKdV equations is proposed in terms of a single Grassmann parameter. Such solutions are shown to satisfy two copies of N=1 supersymmetric mKdV equations connected by nontrivial algebraic identities. Using the super Miura transformation, we obtain solutions of the N=2 super KdV equations. These are shown to generalize solutions derived previously. By using the mKdV/sinh-Gordon hierarchy properties we generate the solutions of the N=2 super sinh-Gordon as well.
In this paper, we investigate the anomalous dispersive relations, inverse scattering transform with a Riemann-Hilbert (RH) problem, and fractional multi-solitons of the integrable combined fractional higher-order mKdV (fhmKdV) hierarchy, including the fractional mKdV (fmKdV), fractional fifth-order mKdV (f5mKdV), fractional combined third-fifth-order mKdV (f35mKdV) equations, etc., which can be featured via completeness of squared scalar eigenfunctions of the ZS spectral problem. We construct a matrix RH problem to present three types of fractional N-solitons illustrating anomalous dispersions…
We discuss some algebraic aspects of the integrable vector non-linear Schrödinger hierarchies (GNLS$_{r}$). These are hierarchies of zero-curvature equations constructed from affine Kac-Moody algebras $\hat{sl}_{r+1}$. Using the dressing transformation method and the tau-function formalism, we construct the N-soliton solutions of the GNLS$_{r}$ systems. The explicit matrix elements in the case of GNLS$_{1}$ are computed using level one vertex operator representations.
A linear system, which generates a Moyal-deformed two-dimensional soliton equation as integrability condition, can be extended to a three-dimensional linear system, treating the deformation parameter as an additional coordinate. The supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter, the flow of which commutes with the flow of the deformed soliton equation. In this way, a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations. We prove the extended hierarch…
In this paper, a new generalized $5\times5$ matrix spectral problem of Ablowitz-Kaup-Newell-Segur(AKNS) type associated with the enlarged matrix Lie super algebra is proposed and its corresponding super soliton hierarchy is established. The super variational identities is used to furnish super-Hamiltonian structures for the resulting super soliton hierarchy.
We study the negative flows of the hierarchy of the integrable Heisenberg Ferromagnet model and their soliton solutions. The first negative flow is related to the so-called short pulse equation. We provide a framework which generates Lax pairs for the other members of the hierarchy. The Inverse scattering, based on the dressing method is illustrated with the derivation of the one-soliton solution.
A new transformation $u=4 ({\rm ln}f)_x$ that can formulate a quintic linear equation and a pair of Hirota's bilinear equations for the (2+1)-dimensional Sawada-Kotera (2DSK) equation is reported firstly, which enables one to obtain a new hierarchy of multiple soliton solutions of the 2DSK equation. It tells a crucial fact that a nonlinear partial differential equation could possess two hierarchies of multiple soliton solutions and the 2DSK equation is the first and only one found in this paper. The quintic linear equation is solved by a pair of Hirota's bilinear equations, of which one is the…
We introduce and study an integrable boundary flow possessing an infinite number of conserving charges which can be thought of as quantum counterparts of the Ablowitz, Kaup, Newell and Segur Hamiltonians. We propose an exact expression for overlap amplitudes of the boundary state with all primary states in terms of solutions of certain ordinary linear differential equation. The boundary flow is terminated at a nontrivial infrared fixed point. We identify a form of whole boundary state corresponding to this fixed point.
We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D^(1)_4 by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.
From the constrained discrete KP (cdKP) hierarchy, the Ablowitz-Ladik lattice has been derived. By means of the gauge transformation, the Wronskian solution of the Ablowitz-Ladik lattice have been given. The $u_1$ of the cdKP hierarchy is a Y-type soliton solution for odd times of the gauge transformation, but it becomes a dark-bright soliton solution for even times of the gauge transformation. The role of the discrete variable $n$ in the profile of the $u_1$ is discussed.
It is known that soliton solutions of the KP-hierarchy corresponds to singular rational curves with only ordinary double points. In this paper we study the degeneration of theta function solutions corresponding to certain trigonal curves. We show that, when the curves degenerate to singular rational curves with only ordinary triple points, the solutions tend to some intermediate solutions between solitons and rational solutions. They are considered as cerain limits of solitons. The Sato Grassmannian is extensively used here to study the degeneration of solutions, since it directly connects sol…
In this article, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-Bäcklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the soliton solutions of the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found.
Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hier…
In this letter, we present a simple and new idea to generate two types of novel integrable multi-Lévy-index and mix-Lévy-index (mixed) fractional nonlinear soliton hierarchies, containing multi-index and mixed fractional higher-order nonlinear Schrödinger (NLS) hierarchy, fractional complex modified Korteweg-de Vries (cmKdV) hierarchy, and fractional mKdV hierarchy. Their explicit forms can be given using the completeness of squared eigenfunctions. Moreover, we present their anomalous dispersion relations via their linearizations, and fractional multi-soliton solutions via the inverse scatteri…
We present the hierarchy and soliton solutions associated to a multi-component generalisation of the modified Korteweg-de Vries equation. A recursive formula for obtaining the Lax operators associated to the higher flows of the hierarchy is provided. Using the method of rational dressing and the symmetries of the Lax operators we obtain the one-soliton solution. We also derive the general rank one-breather solution and express it in terms of determinants. We present the simplest case of the one-breather solution, which is parametrised by two constant unit vectors normal to each other. Finally,…
With the extended logarithmic flow equations and some extended Vertex operators in generalized Hirota bilinear equations, extended bigraded Toda hierarchy(EBTH) was proved to govern the Gromov-Witten theory of orbiford $c_{NM}$ in literature. The generating function of these Gromov-Witten invariants is one special solution of the EBTH. In this paper, the multi-fold Darboux transformations and their determinant representations of the EBTH are given with two different gauge transformation operators. The two Darboux transformations in different directions are used to generate new solutions from k…
Some aspects of the multidimensional soliton geometry are considered. It is shown that some simples (2+1)-dimensional equations are exact reductions of the Self-Dual Yang-Mills equation or its higher hierarchy.
In this paper, we construct a new integrable equation which is a generalization of $q$-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the generalized $q$-Toda equation and a whole integrable generalized $q$-Toda hierarchy are also constructed. To show the integrability, the Bi-hamiltonian structure and tau symmetry of the generalized $q$-Toda hierarchy are given and this leads to the tau function.
We continue the study of the B-Toda hierarchy (the Toda lattice with the constraint of type B) which can be regarded as a discretization of the BKP hierarchy. We introduce the tau-function of the B-Toda hierarchy and obtain the bilinear equations for it. Examples of soliton tau-functions are presented in the explicit form.
A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space $HP^n$. The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces $M=G/H$ by viewing $HP^n \simeq {\rm U}(n+1,H)/{\rm U}(1,H) \times {\rm U}(n,H)\simeq {\rm Sp}(n+1)/{\rm Sp}(1)\times {\rm Sp}(n)$ as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar-vector (multi-component) v…
We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Sc…
Elliptic soliton solutions, i.e., a hierarchy of functions based on an elliptic seed solution, are constructed using an elliptic Cauchy kernel, for integrable lattice equations of Kadomtsev-Petviashvili (KP) type. This comprises the lattice KP, modified KP (mKP) and Schwarzian KP (SKP) equations as well as Hirota's bilinear KP equation, and their successive continuum limits. The reduction to the elliptic soliton solutions of KdV type lattice equations is also discussed.
We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg-de Vries equation and other soliton equations. This is achieved by means of a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev-Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such system the HD-KP hierarchy. Then, we construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD-KP hierarchy. Its role is analogous to the one played by the …
We establish a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors. $τ$ functions in Hirota's form are derived and vertex operators that generate such $τ$ functions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational func…
The Landau-Lifshitz equation is an example of soliton equations with a zero-curvature representation defined on an elliptic curve. This equation can be embedded into an integrable hierarchy of evolution equations called the Landau-Lifshitz hierarchy. This paper elucidates its status in Sato, Segal and Wilson's universal description of soliton equations in the language of an infinite dimensional Grassmann variety. To this end, a Grassmann variety is constructed from a vector space of $2 \times 2$ matrices of Laurent series of the spectral parameter $z$. A special base point $W_0$, called ``vacu…
The generating function of cubic Hodge integrals satisfying the local Calabi-Yau condition is conjectured to be a tau function of a new integrable system which can be regarded as a fractional generalization of the Volterra lattice hierarchy, so we name it the fractional Volterra hierarchy. In this paper, we give the definition of this integrable hierarchy in terms of Lax pair and Hamiltonian formalisms, construct its tau functions, and present its multi-soliton solutions.
N.N. Bogolyubov discovered that the Boltzmann-Enskog kinetic equation has microscopic solutions. They have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these solutions by introduction of a special regularization of the delta-functions. The crucial observation is that the collision integral of the Boltzmann-Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres if th…
From an algebraic construction of the mKdV hierarchy we observe that the space component of the Lax operator play a role of an universal algebraic object. This fact induces the universality of a gauge transformation that relates two field configurations of a given member of the hierarchy. Such gauge transformation generates the Backlund transformation (BT). In this paper we propose a systematic construction of Backlund Transformation for the entire mKdV hierarchy form the known Type-II BT of the sinh-Gordon theory. We explicitly construct the BT of the first few integrable models associated to…
It is by now well known that the wave functions of rational solutions to the KP hierarchy (those which can be achieved as limits of the pure n-soliton solutions) satisfy an additional eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as ``bispectrality'' and has proved to be both interesting and useful. In this note, it is shown that certain (non-rational) soliton solutions of the KP hierarchy satisfy an eigenvalue equation for a non-local operator constructed by composing ordinary differential operators in the spectral parameter with tra…
In this paper, we firstly establish the multi-Hamiltonian structure and infinite many conservation laws for the vector Kaup-Newell hierarchy of the positive and negative orders. The first nontrivial negative flow corresponds to a coupled Fokas-Lenells equation. By constructing a generalized Darboux transformation and using a limiting process, all kinds of one-soliton solutions are constructed including the bright-dark soliton, the dark-anti-dark soliton and the breather-like solutions. Furthermore, multi-bright and multi-dark soliton solutions are derived and their asymptotic behaviors are inv…
New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its Bäcklund transformations
Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT)（圣保罗州立大学理论物理研究所）；School of Mathematics & Hamilton Mathematics Institute, Trinity College Dublin（都柏林三一学院数学与汉密尔顿数学研究所）；Department of Physics, University of Illinois Chicago（伊利诺伊大学芝加哥分校物理系）；Universidade Federal do Espirito Santo, Depto. de Física（埃斯皮里托桑托联邦大学物理系）
This paper is devoted to the negative flows of the AKNS hierarchy. The main result of this work is the functional representation of the extended AKNS hierarchy, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local systems of higher order, derive the generating function of the conservation laws and the N-dark-soliton solutions for the ex…
In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be transformed to the Camassa-Holm equation through a gauge transform. The Lax pair of the equation is derived to guarantee its integrability, and furthermore the equation is shown to have classical solitons, periodic soliton and kink solutions.
We define the set of almost-intertwining matrices to be all triples(X,Y,Z) of n x n matrices for which XZ=YX+T for some rank one matrix T. A surprisingly simple formula is given for tau-functions of the KP hierarchy in terms of such triples. The tau-functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of the matrix X are considered, leading in special cases to the Ruijsenaars-Schneider particle system.
We derive a stochastic AKNS hierarchy using geometrical methods. The integrability is shown via a stochastic zero curvature relation associated with a stochastic isospectral problem. We expose some of the stochastic integrable partial differential equations which extend the stochastic KdV equation discovered by M. Wadati in 1983 for all the AKNS flows. We also show how to find stochastic solitons from the stochastic evolution of the scattering data of the stochastic IST. We finally expose some properties of these equations and also briefly study a stochastic Camassa-Holm equation which reduces…
We investigate bi-Hamiltonian structures and related mKdV hierarchy of solitonic equations generated by (semi) Riemannian metrics and curve flow of non-stretching curves. The corresponding nonholonomic tangent space geometry is defined by canonically induced nonlinear connections, Sasaki type metrics and linear connections. One yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in hierarchies on the tangent bundle described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrodinger map equation.
We study the tau function of the KP-hierarchy associated with an (n,1) curve $y^n=x-α$. If $α=0$ the corresponding tau function is 1. On the other hand if $α\neq 0$ the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex opertaors to the latter we obtain soliton solutions on non-zero constant backgrounds.
We study exact multi-soliton solutions of integrable hierarchies on noncommutative space-times which are represented in terms of quasi-determinants of Wronski matrices by Etingof, Gelfand and Retakh. We analyze the asymptotic behavior of the multi-soliton solutions and found that the asymptotic configurations in soliton scattering process can be all the same as commutative ones, that is, the configuration of N-soliton solution has N isolated localized energy densities and the each solitary wave-packet preserves its shape and velocity in the scattering process. The phase shifts are also the sam…
