Recent zbMATH articles in MSC 37Khttps://www.zbmath.org/atom/cc/37K2021-04-16T16:22:00+00:00WerkzeugIntegrable systems and algebraic geometry. A celebration of Emma Previato's 65th birthday. Volume 1.https://www.zbmath.org/1456.140032021-04-16T16:22:00+00:00"Donagi, Ron (ed.)"https://www.zbmath.org/authors/?q=ai:donagi.ron-y"Shaska, Tony (ed.)"https://www.zbmath.org/authors/?q=ai:shaska.tanushPublisher's description: Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. This first volume covers a wide range of areas related to integrable systems, often emphasizing the deep connections with algebraic geometry. Common themes include theta functions and Abelian varieties, Lax equations, integrable hierarchies, Hamiltonian flows and difference operators. These powerful tools are applied to spinning top, Hitchin, Painlevé and many other notable special equations.
The articles of this volume will be reviewed individually. For Vol. 2 see [Zbl 1456.14004].Quantum Gaudin model, spin chains, and universal characters.https://www.zbmath.org/1456.821482021-04-16T16:22:00+00:00"Li, Chuanzhong"https://www.zbmath.org/authors/?q=ai:li.chuanzhong.1|li.chuanzhong"Shou, Bao"https://www.zbmath.org/authors/?q=ai:shou.baoIn this paper, the authors construct a connection between the quantum Gaudin model [\textit{A. Alexandrov} et al., Nucl. Phys., B 883, 173--223 (2014; Zbl 1323.37037)] and the universal character hierarchy by a newly defined
coupled master \(T\)-operator [\textit{A. Alexandrov} et al., J. High Energy Phys. 2013, No. 9, Paper No. 064, 64 p. (2013; Zbl 1342.81179)]. The coupled master \(T\)-operator of the quantum Gaudin model satisfies the bilinear identity of the universal
character hierarchy, which is an extension of the Kadomtsev-Petviashvili hierarchy [\textit{E. Date} et al., in: Nonlinear integrable systems -- classical theory and quantum theory, Proc. RIMS Symp., Kyoto 1981, 39--119 (1983; Zbl 0571.35098)]. In addition, for the generalized quantum integrable spin
chains with rational \(\mathrm{GL}(N)\)-invariant \(R\)-matrices, the authors construct its coupled master \(T\)-operator, which represents a generating function for
two-folds commuting quantum transfer matrices. The functional relations for the transfer matrices are equivalent to an infinite set of Hirota
bilinear equations of the modified universal character hierarchy.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation.https://www.zbmath.org/1456.350072021-04-16T16:22:00+00:00"He, Xue-Jiao"https://www.zbmath.org/authors/?q=ai:he.xue-jiao"Lü, Xing"https://www.zbmath.org/authors/?q=ai:lu.xing"Li, Meng-Gang"https://www.zbmath.org/authors/?q=ai:li.menggangSummary: With the Hirota bilinear method and symbolic computation, we investigate the \((3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation. Based on its bilinear form, the bilinear Bäcklund transformation is constructed, which consists of four equations and five free parameters. The Pfaffian, Wronskian and Grammian form solutions are derived by using the properties of determinant. As an example, the one-, two- and three-soliton solutions are constructed in the context of the Pfaffian, Wronskian and Grammian forms. Moreover, the triangle function solutions are given based on the Pfaffian form solution. A few particular solutions are plotted by choosing the appropriate parameters.Vertex operators, solvable lattice models and metaplectic Whittaker functions.https://www.zbmath.org/1456.820972021-04-16T16:22:00+00:00"Brubaker, Ben"https://www.zbmath.org/authors/?q=ai:brubaker.ben"Buciumas, Valentin"https://www.zbmath.org/authors/?q=ai:buciumas.valentin"Bump, Daniel"https://www.zbmath.org/authors/?q=ai:bump.daniel"Gustafsson, Henrik P. A."https://www.zbmath.org/authors/?q=ai:gustafsson.henrik-p-aThis paper discusses two mechanisms by which the quantum groups \(U_q (\hat{\mathfrak{g}})\), for a simple Lie algebra or superalgebra \(\mathfrak{g}\), produce families of special functions with a number of interesting properties related to functional equations, branching rules and unexpected algebraic relations. The first mechanism uses solvable lattice models associated to finite-dimensional modules of \(U_q (\hat{\mathfrak{g}})\). The second mechanism uses actions of Heisenberg and Clifford algebras on a fermionic Fock space, exploiting the boson-fermion correspondence arising in connection with soliton theory, dating back to [\textit{M. Jimbo} and \textit{T. Miwa}, Publ. Res. Inst. Math. Sci. 19, 943--1001 (1983; Zbl 0557.35091)] and pushed forward by \textit{T. Lam} [Math. Res. Lett. 13, No. 2--3, 377--392 (2006; Zbl 1160.05056)] and especially by [\textit{M. Kashiwara} et al., Sel. Math., New Ser. 1, No. 4, 787--805 (1995; Zbl 0857.17013)]. These two points of view provide new insight into the theory of metaplectic Whittaker functions for the general linear group and relate them to LLT polynomials (known also as ribbon symmetric functions). The main theorem of the paper considers two solvable lattice models, named Gamma ice and Delta, and details in Section 4 their row transfer matrices. In this study, metaplectic ice models are exploited, whose partition functions are metaplectic Whittaker functions. In the process, the authors introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions. It is explained that half vertex operators agree with Lam's construction, and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials [\textit{A. Lascoux} et al., J. Math. Phys. 38, No. 2, 1041--1068 (1997; Zbl 0869.05068)] can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models. A number of links with the existing literature is identified as well.
Reviewer: Piotr Garbaczewski (Opole)Primitive solutions of the Korteweg-de Vries equation.https://www.zbmath.org/1456.370712021-04-16T16:22:00+00:00"Dyachenko, S. A."https://www.zbmath.org/authors/?q=ai:dyachenko.sergey-a"Nabelek, P."https://www.zbmath.org/authors/?q=ai:nabelek.p"Zakharov, D. V."https://www.zbmath.org/authors/?q=ai:zakharov.dmitry"Zakharov, V. E."https://www.zbmath.org/authors/?q=ai:zakharov.vladimir-eThe authors of this paper survey recent results
concerning a new family of solutions, called
``primitive solutions'' of the
Korteweg-de Vries (KdV)
equation,
\[
u_t=6uu_x-u_{xxx}.
\]
In this differential
equation \(u=u(x,t)\) is the unknown function.
It is the first equation
of an infinite sequence of
commuting equations called the KdV hierarchy.
The auxiliary linear operator for the KdV
hierarchy is the
one-dimensional Schrödinger operator
defined on the real axis.
The primitive solutions are defined as limits
of rapidly vanishing solutions of the KdV
equation. A primitive
solution is determined nonuniquely by three different
functions. The first two of them are positive
functions on an interval on
the imaginary axis. The third one is
a function on the real axis determining
the reflection coefficient.
It is shown that the
elliptic one-gap solutions
and periodic finite-gap solutions are special
cases of reflectionless primitive solutions.
The authors
study the case of periodic initial data
using algebro-geometric finite-gap solutions.
Such a solution
is determined by a hyperelliptic algebraic curve
with real branch points and a
divisor on it. It turns out that the solution can be
explicitly given by the Matveev-Its formula
in terms of the Riemann theta function of the
spectral curve. It is known that the
periodic finite-gap solutions are dense in the
space of all periodic solutions.
It is known that
periodic finite-gap
solutions of the KdV equation
can be obtained from \(N\)-soliton solutions in
the limit \(N \to \infty \).
However, a precise description
of such a limit was unknown.
Reviewer: Dimitar A. Kolev (Sofia)Reductions of the strict KP hierarchy.https://www.zbmath.org/1456.370732021-04-16T16:22:00+00:00"Helminck, G. F."https://www.zbmath.org/authors/?q=ai:helminck.gerardus-franciscus"Panasenko, E. A."https://www.zbmath.org/authors/?q=ai:panasenko.elena-aSummary: Let \(R\) be a commutative complex algebra and \(\partial\) be a \(\mathbb{C} \)-linear derivation of \(R\) such that all powers of \(\partial\) are \(R\)-linearly independent. Let \(R[ \partial ]\) be the algebra of differential operators in \(\partial\) with coefficients in \(R\) and \(Psd\) be its extension by the pseudodifferential operators in \(\partial\) with coefficients in \(R\). In the algebra \(R[ \partial ]\), we seek monic differential operators \(\mathbf{M}_n\) of order \(n\ge2\) without a constant term satisfying a system of Lax equations determined by the decomposition of \(Psd\) into a direct sum of two Lie algebras that lies at the basis of the strict KP hierarchy. Because this set of Lax equations is an analogue for this decomposition of the \(n\)-KdV hierarchy, we call it the strict \(n\)-KdV hierarchy. The system has a minimal realization, which allows showing that it has homogeneity properties. Moreover, we show that the system is compatible, i.e., the strict differential parts of the powers of \(M=( \mathbf{M}_n)^{1/n}\) satisfy zero-curvature conditions, which suffice for obtaining the Lax equations for \(\mathbf{M}_n\) and, in particular, for proving that the \(n\)th root \(M\) of \(\mathbf{M}_n\) is a solution of the strict KP theory if and only if \(\mathbf{M}_n\) is a solution of the strict \(n\)-KdV hierarchy. We characterize the place of solutions of the strict \(n\)-KdV hierarchy among previously known solutions of the strict KP hierarchy.Virasoro symmetries of multicomponent Gelfand-Dickey systems.https://www.zbmath.org/1456.370702021-04-16T16:22:00+00:00"An, Ling"https://www.zbmath.org/authors/?q=ai:an.ling"Li, Chuanzhong"https://www.zbmath.org/authors/?q=ai:li.chuanzhong.1|li.chuanzhongSummary: We study the additional symmetries and \(\tau \)-functions of multicomponent Gelfand-Dickey hierarchies, which include classical integrable systems such as the multicomponent Korteweg-de Vries and Boussinesq hierarchies. Using various reductions, we derive B- and C-type multicomponent Gelfand-Dickey hierarchies. We show that not all flows of their additional symmetries survive. We find that the generators of the additional symmetries of the B- and C-type multicomponent Gelfand-Dickey hierarchies differ but the forms of their additional flows are the same.Two-component generalized Ragnisco-Tu equation and the Riemann-Hilbert problem.https://www.zbmath.org/1456.370762021-04-16T16:22:00+00:00"Wang, Linlin"https://www.zbmath.org/authors/?q=ai:wang.linlin"Song, Caiqin"https://www.zbmath.org/authors/?q=ai:song.caiqin"Zhu, Junyi"https://www.zbmath.org/authors/?q=ai:zhu.junyiSummary: Using the Riemann-Hilbert approach, we investigate the two-component generalized Ragnisco-Tu equation. The modified equation is integrable in the sense that a Lax pair exists, but its explicit solutions have some distinctive properties. We show that the explicit one-wave solution is unstable and the two-wave solution preserves only the phase shift but not the wave shape after collision.Spectral stability of nonlinear waves in KdV-type evolution equations.https://www.zbmath.org/1456.370792021-04-16T16:22:00+00:00"Pelinovsky, Dmitry E."https://www.zbmath.org/authors/?q=ai:pelinovsky.dmitry-eAuthor's abstract: This chapter focuses on the spectral stability of nonlinear waves in Korteweg-de Vries (KdV) type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and an unbounded non-invertible operator \(\partial_x\). The instability index theorem is proven under a generic assumption on the self-adjoint operator both in the case of solitary waves and periodic waves. This result is reviewed in the context of recent results on spectral stability of nonlinear waves in KdV-type evolution equations.
For the entire collection see [Zbl 1280.37002].
Reviewer: Kaïs Ammari (Monastir)Elliptic solutions of the semidiscrete B-version of the Kadomtsev-Petviashvili equation.https://www.zbmath.org/1456.370752021-04-16T16:22:00+00:00"Rudneva, D. S."https://www.zbmath.org/authors/?q=ai:rudneva.d-s"Zabrodin, A. V."https://www.zbmath.org/authors/?q=ai:zabrodin.anton-vSummary: We study elliptic solutions of the semidiscrete B-version of the Kadomtsev-Petviashvili equation and derive the equations of motion of their poles. The auxiliary linear problems for the wave function are the main technical tool.Constructing a quantum Lax pair from Yang-Baxter equations.https://www.zbmath.org/1456.812432021-04-16T16:22:00+00:00"Lima-Santos, A."https://www.zbmath.org/authors/?q=ai:lima-santos.antonioIntegrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws.https://www.zbmath.org/1456.350732021-04-16T16:22:00+00:00"Lü, Xing"https://www.zbmath.org/authors/?q=ai:lu.xing"Hua, Yan-Fei"https://www.zbmath.org/authors/?q=ai:hua.yan-fei"Chen, Si-Jia"https://www.zbmath.org/authors/?q=ai:chen.sijia"Tang, Xian-Feng"https://www.zbmath.org/authors/?q=ai:tang.xian-fengSummary: The (2+1)-dimensional Kadomtsev-Petviashvili type equations describe the nonlinear phenomena and characteristics in oceanography, fluid dynamics and shallow water. In the literature, a novel (2+1)-dimensional nonlinear model is proposed, and the localized wave interaction solutions are studied including lump-kink and lump-soliton types. Hereby, it is of further value to investigate the integrability characteristics of this model. In this paper, we firstly conduct the Painlevé analysis and find it fails to pass the Painlevé test due to a non-vanishing compatibility condition at the highest resonance level. Then we derive the soliton solutions and give the formula of the \(N\)-soliton solution, which is proved by means of the Hirota condition. The criterion for the linear superposition principle is also given to generate the resonant solutions. Bäcklund transformation, Lax pair and infinitely many conservation laws are derived through the Hirota bilinear method and Bell polynomial approach. As a result, we have a more overall understanding of the integrability characteristics of this novel (2+1)-dimensional nonlinear model.Recursion operators and hierarchies of mKdV equations related to the Kac-Moody algebras \(D_4^{(1)}\), \(D_4^{(2)}\), and \(D_4^{(3)}\).https://www.zbmath.org/1456.370722021-04-16T16:22:00+00:00"Gerdjikov, V. S."https://www.zbmath.org/authors/?q=ai:gerdzhikov.vladimir-stefanov"Stefanov, A. A."https://www.zbmath.org/authors/?q=ai:stefanov.aleksander-a"Iliev, I. D."https://www.zbmath.org/authors/?q=ai:iliev.ilya-d|iliev.iliya-dimov|iliev.ilija-d"Boyadjiev, G. P."https://www.zbmath.org/authors/?q=ai:boyadjiev.g-p"Smirnov, A. O."https://www.zbmath.org/authors/?q=ai:smirnov.alexander-o"Matveev, V. B."https://www.zbmath.org/authors/?q=ai:matveev.vladimir-b"Pavlov, M. V."https://www.zbmath.org/authors/?q=ai:pavlov.maxim-vSummary: We construct three nonequivalent gradings in the algebra \(D_4\simeq so(8)\). The first is the standard grading obtained with the Coxeter automorphism \(C_1=S_{\alpha_2}S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}\) using its dihedral realization. In the second, we use \(C_2=C_1R\), where \(R\) is the mirror automorphism. The third is \(C_3=S_{\alpha_2}S_{\alpha_1}T\), where \(T\) is the external automorphism of order 3. For each of these gradings, we construct a basis in the corresponding linear subspaces \(\mathfrak{g}^{(k)} \), the orbits of the Coxeter automorphisms, and the related Lax pairs generating the corresponding modified Korteweg-de Vries (mKdV) hierarchies. We find compact expressions for each of the hierarchies in terms of recursion operators. Finally, we write the first nontrivial mKdV equations and their Hamiltonians in explicit form. For \(D_4^{(1)} \), these are in fact two mKdV systems because the exponent 3 has the multiplicity two in this case. Each of these mKdV systems consists of four equations of third order in \(\partial_x\). For \(D_4^{(2)} \), we have a system of three equations of third order in \(\partial_x\). For \(D_4^{(3)}\), we have a system of two equations of fifth order in \(\partial_x\).On the integrability of lattice equations with two continuum limits.https://www.zbmath.org/1456.370822021-04-16T16:22:00+00:00"Garifullin, R. N."https://www.zbmath.org/authors/?q=ai:garifullin.rustem-n|garifullin.rustem-nailevich"Yamilov, R. I."https://www.zbmath.org/authors/?q=ai:yamilov.ravil-islamovichSummary: We study a new example of a lattice equation, which is one of the key equations of a generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits, which are the well-known fifth-order partial-differential equations, namely, the Sawada-Kotera and Kaup-Kupershmidt equations. We justify its integrability by constructing an \(L-A\) pair and a hierarchy of conservation laws.Symmetry drivers and formal integrals of hyperbolic systems.https://www.zbmath.org/1456.370682021-04-16T16:22:00+00:00"Startsev, S. Ya."https://www.zbmath.org/authors/?q=ai:startsev.sergey-ya|startsev.sergei-yakovlevichSummary: In this paper, we consider symmetry drivers (i.e., operators that map arbitrary functions of one of independent variables into symmetries) and formal integrals (i.e., operators that map symmetries to the kernel of the total derivative). We prove that a hyperbolic system of partial differential equations possesses a complete set of formal integrals if and only if it admits a complete set of symmetry drivers. This assertion is also valid for difference and differential-difference analogs of scalar hyperbolic equations.Phase model expectation values and the 2-Toda hierarchy.https://www.zbmath.org/1456.823302021-04-16T16:22:00+00:00"Zuparic, M."https://www.zbmath.org/authors/?q=ai:zuparic.mathew-lBifurcations and exact traveling wave solutions of two shallow water two-component systems.https://www.zbmath.org/1456.351632021-04-16T16:22:00+00:00"Li, Jibin"https://www.zbmath.org/authors/?q=ai:li.jibin"Chen, Guanrong"https://www.zbmath.org/authors/?q=ai:chen.guanrong"Zhou, Yan"https://www.zbmath.org/authors/?q=ai:zhou.yanGaussian solitary waves for argument-Schrödinger equation.https://www.zbmath.org/1456.370772021-04-16T16:22:00+00:00"Yamano, Takuya"https://www.zbmath.org/authors/?q=ai:yamano.takuyaSummary: We present localized analytical solutions of the logarithmic nonlinear Schrödinger equation, i.e., the so-called the argument-Schrödinger equation. The Gaussian solitary waveform is shown to be the solution, and we obtain the explicit form in a one-dimensional case when the dynamics evolve under a quadratic potential. The dispersion relation becomes time-dependent due to the logarithmic nonlinearity.Nonlinear flag manifolds as coadjoint orbits.https://www.zbmath.org/1456.370602021-04-16T16:22:00+00:00"Haller, Stefan"https://www.zbmath.org/authors/?q=ai:haller.stefan"Vizman, Cornelia"https://www.zbmath.org/authors/?q=ai:vizman.corneliaIn [Math. Ann. 329, No. 4, 771--785 (2004; Zbl 1071.58005)], the present authors introduced the notion of a nonlinear Grassmannian and studied the Fréchet manifold \(\mathrm{Gras}_n(M)\) of all \(n\)-dimensional oriented compact submanifolds of a smooth closed connected \(m\)-dimensional manifold \(M\).
They showed that every closed \((n+2)\)-form \(\alpha\) on \(M\) defines a closed 2-form \(\widetilde{\alpha}\) on \(\mathrm{Gras}_n(M)\), and if \(\alpha\) is integrable, then \(\widetilde{\alpha}\) is the curvature form of a principal connection on a principal \(S^1\)-bundle over \(\mathrm{Gras}_n(M)\). In the case \(\alpha\) is a closed, integrable volume form, then every connected component \(\mathcal{M}\) of \(\mathrm{Gras}_{m-2}(M)\), equipped with the symplectic form \(\widetilde{\alpha}\), is a prequantizable coadjoint orbit of some central extension of the Hamiltonian group \(\text{Ham}(M,\alpha)\) by \(S^1\).
In this paper, the authors generalize the notion of a nonlinear Grassmannian to the notion of a nonlinear flag manifold.
If \(M\) is a smooth manifold, \(S_1,\dots,S_r\) are closed smooth manifolds, then a sequence of nested embedded submanifolds \(N_1\subseteq\dots\subseteq N_r\subseteq M\) such that \(N_i\) is diffeomorphic to \(S_i\) for all \(i=1,\dots,r\) is called a nonlinear flag of type \(\mathscr{S}=(S_1,\dots,S_r)\) in \(M\).
The space of all nonlinear flags of type \(\mathscr{S}\) in \(M\) can be equipped with the structure of a Fréchet manifold in a natural way and is denoted by \(\mathrm{Flag}_{\mathscr{S}}(M)\).
The main goal of this paper is to study the geometry of this space.
A nonlinear Grassmannian is a special case of a nonlinear flag and corresponds to the case \(r=1\).
The authors present some applications of nonlinear flag manifolds by using them to describe certain coadjoint orbits of the Hamiltonian group.
If \(M\) is a closed symplectic manifold, \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) is the open subset in \(\mathrm{Flag}_{\mathscr{S}}(M)\) consisting of all symplectic flags of type \(\mathscr{S}\), then the symplectic form on \(M\) induces by transgression a symplectic form on the manifold of symplectic nonlinear flags. The Hamiltonian group \(\mathrm{Ham}(M)\) acts on \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) in a Hamiltonian fashion with equivariant moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\).
This moment map is injective and identifies each connected component of \(\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\) with a coadjoint orbit of \(\mathrm{Ham}(M)\).
The main result of the paper states that the restriction of the moment map \(J:\mathrm{Flag}_{\mathscr{S}}^{\mathrm{symp}}(M)\to\mathfrak{ham}(M)^*\) to any connected component is one-to-one onto a coadjoint orbit cf the Hamiltonian group \(\mathrm{Ham}(M)\). The Kostant-Kirillov-Souriau symplectic form \(\omega_{\mathrm{KKS}}\) on the coadjoint orbit satisfies \(J^*\omega_{\mathrm{KKS}}=\Omega\), where \(\Omega\) is a natural symplectic form.
Reviewer: Andrew Bucki (Edmond)Bäcklund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota-Satsuma-Ito-like equation.https://www.zbmath.org/1456.351782021-04-16T16:22:00+00:00"Chen, Si-Jia"https://www.zbmath.org/authors/?q=ai:chen.sijia"Ma, Wen-Xiu"https://www.zbmath.org/authors/?q=ai:ma.wen-xiu"Lü, Xing"https://www.zbmath.org/authors/?q=ai:lu.xingThe paper aims to construct exact solutions to the (3 + 1)-dimensional system of the Hirota-Satsuma-Ito-type, as a generalization of the known (2 + 1)-dimensional Hirota-Satsuma-Ito equation. The solutions are produced using a possibility to cast the system in the form of the Hirota bilinear represenation. In particular, the Bäcklund transformation is derived for the system, starting from the Hirota method. The analysis is focused on solutions in the form of lumps, i.e., solitons with a dipole structure. Further, solutions are constructed for interaction of lumps with kinks and multi-kink complexes. The interactions may be both elastic and inelastic. In the latter case, the original lump is absorbed by the kinks, which is a nontrivial example of strongly inelastic interactions in the integrable system.
Reviewer: Boris A. Malomed (Tel Aviv)Generalized symplectic Schur functions and SUC hierarchy.https://www.zbmath.org/1456.370742021-04-16T16:22:00+00:00"Huang, Fang"https://www.zbmath.org/authors/?q=ai:huang.fang"Wang, Na"https://www.zbmath.org/authors/?q=ai:wang.naThe authors define a generalization of
symplectic Schur functions and their vertex operator realizations.
They obtain a series of
nonlinear partial differential equations of infinite order, called
symplectic universal character hierarchy; they regard it as an extension of the symplectic
Kadomtsev-Petviashvili (KP) hierarchy.
This paper is organized as
follows. Section 1 is an introduction to the subject. In Sections
2 and 3, the authors recall some results on Schur functions in general
and in relation with solutions of the KP hierarchy. In Section 3,
they define an integrable system whose tau function can be
obtained from the symplectic Schur function. In Section 4, the
authors define the generalized symplectic Schur function, which is
an extension of the symplectic Schur function and construct its
vertex operator realization. They prove that these functions can be identified as
vacuum expectation values of fermionic operators. They give
the boson-fermion correspondence for the generalized symplectic
Schur functions and construct a modified integrable
system, thus proving that the generalized symplectic Schur functions
are solutions of it.
Reviewer: Ahmed Lesfari (El Jadida)Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies.https://www.zbmath.org/1456.370692021-04-16T16:22:00+00:00"Álvarez, Gabriel"https://www.zbmath.org/authors/?q=ai:alvarez.gabriel"Martínez Alonso, Luis"https://www.zbmath.org/authors/?q=ai:martinez-alonso.luis"Medina, Elena"https://www.zbmath.org/authors/?q=ai:medina.elenaModulation of solitary waves and formation of stable attractors in granular scalar models subjected to on-site perturbation.https://www.zbmath.org/1456.740972021-04-16T16:22:00+00:00"Ben-Meir, Y."https://www.zbmath.org/authors/?q=ai:ben-meir.y"Starosvetsky, Y."https://www.zbmath.org/authors/?q=ai:starosvetsky.yuliSummary: Present work concerns the propagation of solitary waves in the array of coupled, uncompressed granular chains subjected to onsite perturbation. We devise a special analytical procedure depicting the modulation of solitary pulses caused by the inter-chain interaction as well as by the on-site perturbations of a general type. The proposed analytical procedure is very efficient in depicting both the transient response characterized by significant energy fluctuations between the chains as well as in predicting the formation of stable attractors corresponding to a steady state response. We confirm the validity of a general analytical procedure with several specific setups of granular scalar models. In particular we consider the response of the array of coupled granular chains free of perturbation as well as the arrays subject to the basic type of on-site perturbations such as the ones induced by the uniform and random elastic foundation, dissipation. Additional interesting finding made in the present study corresponds to the granular arrays subject to a special type of on-site perturbation containing the terms leading to the two opposing effects namely dissipation and energy sourcing. Interestingly enough this type of perturbation may lead to the formation of stable attractors. By the term attractors we refer to the stable, stationary pulses simultaneously forming on all the coupled chains and propagating with the same phase speed. It is important to emphasize that the analytical procedure developed in the first part of the study predicts the formation of stable attractors through a typical saddle-node bifurcation. Moreover, results of the reduced model are found to be in a spectacular agreement with those of the direct numerical simulations of the true model.Quasi-periodic solutions for fractional nonlinear Schrödinger equation.https://www.zbmath.org/1456.370802021-04-16T16:22:00+00:00"Xu, Xindong"https://www.zbmath.org/authors/?q=ai:xu.xindongSummary: We establish an infinite dimensional KAM theorem with dense normal frequency. As an application, we use this theorem to study the fractional NLS
\[
iu_t-|\partial_x|^{\frac{1}{2}}u+M_\xi u=\varepsilon f(|u|^2)u,\quad x\in\mathbb{T},t\in\mathbb{R},
\]
where \(f\) is real analytic in a neighborhood of \(0\in \mathbb{C}\) and \(\varepsilon >0\) is small enough. We obtain a family of small-amplitude quasi-periodic solutions with linear stability.Bifurcations of traveling wave solutions for a generalized Camassa-Holm equation.https://www.zbmath.org/1456.350752021-04-16T16:22:00+00:00"Wei, Minzhi"https://www.zbmath.org/authors/?q=ai:wei.minzhi"Sun, Xianbo"https://www.zbmath.org/authors/?q=ai:sun.xianbo"Zhu, Hongying"https://www.zbmath.org/authors/?q=ai:zhu.hongyingSummary: In this paper, the traveling wave solutions for a generalized Camassa-Holm equation \(u_t-u_{xxt}=\frac{1}{2}(p+1)(p+2)u^pu_x-\frac{1}{2}p(p-1)u^{p-2}u_x^3-2pu^{p-1}u_xu_{xx}-u^pu_{xxx}\) are investigated. By using the bifurcation method of dynamical systems, three major results for this equation are highlighted. First, there are one or two singular straight lines in the two-dimensional system under some different conditions. Second, all the bifurcations of the generalized Camassa-Holm equation are given for \(p\) either positive or negative integer. Third, we prove that the corresponding traveling wave system of this equation possesses peakon, smooth solitary wave solution, kink and anti-kink wave solution, and periodic wave solutions.Domain wall partition functions and KP.https://www.zbmath.org/1456.822582021-04-16T16:22:00+00:00"Foda, O."https://www.zbmath.org/authors/?q=ai:foda.omar"Wheeler, M."https://www.zbmath.org/authors/?q=ai:wheeler.michael"Zuparic, M."https://www.zbmath.org/authors/?q=ai:zuparic.mathew-lPainlevé test, complete symmetry classifications and exact solutions to R-D types of equations.https://www.zbmath.org/1456.370832021-04-16T16:22:00+00:00"Liu, Hanze"https://www.zbmath.org/authors/?q=ai:liu.hanze"Bai, Cheng-Lin"https://www.zbmath.org/authors/?q=ai:bai.chenglin"Xin, Xiangpeng"https://www.zbmath.org/authors/?q=ai:xin.xiangpengSummary: In this paper, the combination of Painlevé analysis and symmetry classification is performed on the reaction-diffusion (R-D) types of equations, the Painlevé properties (PPs) and Bäcklund transformations (BTs) are obtained under some conditions. Then, all of the point symmetries of the equations are given by Lie group classification method, moreover, the complete generalized symmetry classifications of the general R-D equation are provided using characteristics of predetermined order, and the integrability of the nonlinear equations are considered by the generalized symmetry classification method. Furthermore, the exact solutions to the equations generated from Painlevé expansions and symmetry reductions are investigated.Quasi-periodic solutions of PDEs.https://www.zbmath.org/1456.350102021-04-16T16:22:00+00:00"Berti, Massimiliano"https://www.zbmath.org/authors/?q=ai:berti.massimilianoSummary: The aim of this talk is to present some recent existence results about quasi-periodic solutions for PDEs like nonlinear wave and Schrödinger equations in \(\mathbb{T}^d\),\(d\geq 2\), and the 1-d derivative wave equation. The proofs are based on both Nash-Moser implicit function theorems and KAM theory.Investigation of solitons and mixed lump wave solutions with \((3+1)\)-dimensional potential-YTSF equation.https://www.zbmath.org/1456.370782021-04-16T16:22:00+00:00"Younis, Muhammad"https://www.zbmath.org/authors/?q=ai:younis.muhammad"Ali, Safdar"https://www.zbmath.org/authors/?q=ai:ali.safdar"Rizvi, Syed Tahir Raza"https://www.zbmath.org/authors/?q=ai:rizvi.syed-tahir-raza"Tantawy, Mohammad"https://www.zbmath.org/authors/?q=ai:tantawy.mohammad"Tariq, Kalim U."https://www.zbmath.org/authors/?q=ai:tariq.kalim-u"Bekir, Ahmet"https://www.zbmath.org/authors/?q=ai:bekir.ahmetSummary: The article investigates the exact and mixed lump wave solitons to the \((3+1)\)-dimensional potential YTSF equation, which is an extension of the Bogoyavlenskii-Schif equation. Different types of interaction solutions in terms of a new merge of positive quadratic functions, trigonometric functions and hyperbolic functions are obtained, which are investigated using the extended three soliton test approach. The dynamical behavior of solutions has also depicted in different 3D representations. The representations show that the soliton solutions are obtained in the form of train and cusp. Some other solutions like lamb wave with high peak, and lump solutions with kink background are also observed in these representations.Properties of some breather solutions of a nonlocal discrete NLS equation.https://www.zbmath.org/1456.370812021-04-16T16:22:00+00:00"Ben, Roberto I."https://www.zbmath.org/authors/?q=ai:ben.roberto-i"Borgna, Juan Pablo"https://www.zbmath.org/authors/?q=ai:borgna.juan-pablo"Panayotaros, Panayotis"https://www.zbmath.org/authors/?q=ai:panayotaros.panayotisSummary: We present results on breather solutions of a discrete nonlinear Schrödinger equation with a cubic Hartree-type nonlinearity that models laser light propagation in waveguide arrays that use a nematic liquid crystal substratum. A recent study of that model by \textit{R. I. Ben} et al. [Phys. Lett., A 379, No. 30--31, 1705--1714 (2015; Zbl 1343.35210)] showed that nonlocality leads to some novel properties such as the existence of orbitaly stable breathers with internal modes, and of shelf-like configurations with maxima at the interface. In this work, we present rigorous results on these phenomena and consider some more general solutions. First, we study energy minimizing breathers, showing existence as well as symmetry and monotonicity properties. We also prove results on the spectrum of the linearization around one-peak breathers, solutions that are expected to coincide with minimizers in the regime of small linear intersite coupling. A second set of results concerns shelf-type breather solutions that may be thought of as limits of solutions examined in [\textit{R. I. Ben} et al., Phys. Lett., A 379, No. 30--31, 1705--1714 (2015; Zbl 1343.35210)]. We show the existence of solutions with a non-monotonic front-like shape and justify computations of the essential spectrum of the linearization around these solutions in the local and nonlocal cases.Integrable systems and algebraic geometry. A celebration of Emma Previato's 65th birthday. Volume 2.https://www.zbmath.org/1456.140042021-04-16T16:22:00+00:00"Donagi, Ron (ed.)"https://www.zbmath.org/authors/?q=ai:donagi.ron-y"Shaska, Tony (ed.)"https://www.zbmath.org/authors/?q=ai:shaska.tanushPublisher's description: Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. The articles in this second volume discuss areas related to algebraic geometry, emphasizing the connections of this central subject to integrable systems, arithmetic geometry, Riemann surfaces, coding theory and lattice theory.
The articles of this volume will be reviewed individually. For Vol. 1 see [Zbl 1456.14003].Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients.https://www.zbmath.org/1456.350722021-04-16T16:22:00+00:00"Chen, Si-Jia"https://www.zbmath.org/authors/?q=ai:chen.sijia"Lü, Xing"https://www.zbmath.org/authors/?q=ai:lu.xing"Tang, Xian-Feng"https://www.zbmath.org/authors/?q=ai:tang.xian-fengSummary: A generalized Burgers equation with variable coefficients is introduced based on the (2+1)-dimensional Burgers equation. Using the test function method combined with the bilinear form, we obtain the lump solutions to the generalized Burgers equation with variable coefficients. The amplitude and velocity of the extremum point are derived to analyze the propagation of the lump wave. Moreover, we derive and study the mixed solutions including lump-one-kink and lump-two-kink cases. With symbolic computation, two cases of relations among the parameters are yielded corresponding to the solutions. Different and interesting interaction phenomena arise from assigning abundant functions to the variable coefficients. Especially, we find that the shape of kink waves might be parabolic type, and one lump wave can be decomposed into two lump waves. The test function method is applicable for the generalized Burgers equation with variable coefficients, and it will be applied to some other variable-coefficient equations in the future.Hopf bifurcation of KdV-Burgers-Kuramoto system with delay feedback.https://www.zbmath.org/1456.371072021-04-16T16:22:00+00:00"Guan, Junbiao"https://www.zbmath.org/authors/?q=ai:guan.junbiao"Liu, Jie"https://www.zbmath.org/authors/?q=ai:liu.jie|liu.jie.3|liu.jie.4|liu.jie.5|liu.jie.2|liu.jie.1|liu.jie.7"Feng, Zhaosheng"https://www.zbmath.org/authors/?q=ai:feng.zhaoshengDegeneration of breathers in the Kadomttsev-Petviashvili I equation.https://www.zbmath.org/1456.351832021-04-16T16:22:00+00:00"Yuan, Feng"https://www.zbmath.org/authors/?q=ai:yuan.feng"Cheng, Yi"https://www.zbmath.org/authors/?q=ai:cheng.yi"He, Jingsong"https://www.zbmath.org/authors/?q=ai:he.jingsongThe paper reports exact solutions for higher-order breathers of the Kadomttsev-Petviashvili I equation. The solutions are produced by means of the Hirota bilinear method and the complexication method. A relation between such solutions and famous lump solitons produced by this equation is analyzed. Also constructed are hybrid states coimposed of solitons, breathers, and lumps.
Reviewer: Boris A. Malomed (Tel Aviv)