Dimensional Integrable Systems Research Articles

Integrable Systems of the Intermediate Long Wave Type in 2 + 1 Dimensions

We classify 2+1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to the 2+1 dimensional waterbag system are established. Dimensional reductions of integrable systems constructed in this paper provide dispersive regularisations of hydrodynamic equations governing propagation of long nonlinear waves in a shear flow with piecewise linear velocity profile (for special values of vorticities).

Interaction of high-order breather, periodic wave, lump, rational soliton solutions and mixed solutions for reductions of the (4+1)-dimensional Fokas equation

The interaction of high-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions for reductions of the (4+1)-dimensional Fokas equation are investigated by means of the Kadomtsev-Petviashvili (KP) hierarchy reduction method. Through analyzing the structural characteristics of periodic wave solutions, we find that evolution of the breather is decided by two characteristic lines. Interestingly, growing-decaying amplitude periodic wave and amplitude-invariant periodic wave are given through some conditions posed on the parameters. Some fascinating nonlinear wave patterns composed of high-order breathers and high-order periodic waves are shown. Furthermore, taking the long wave limit on the periodic-wave solutions, the semi-rational solutions composed of lumps, moving solitons, breathers, and periodic waves are obtained. Some novel dynamical processes are graphically analyzed. Additionally, we provide a new method to derive periodic-wave and semi-rational solutions for the (3+1)-dimensional KP equation by reducing the solutions of the (4+1)-dimensional Fokas equation. The presented results might help to understand the dynamic behaviors of nonlinear waves in the fluid fields and may provide some new perspectives for studying nonlinear wave solutions of high dimensional integrable systems.

$$\bar\partial$$-dressing method for a few $$(2+1)$$-dimensional integrable coupling systems

$$\bar\partial$$-dressing method for a few $$(2+1)$$-dimensional integrable coupling systems

Higher-order rogue wave solutions to the Kadomtsev–Petviashvili 1 equation

We construct a family of eigenfunction solutions of the Lax pair of the Kadomtsev–Petviashvili 1 (KP1) equation, which is expressed in terms of the Taylor coefficients of a fundamental exponential function associated with the Lax pair. Using the binary Darboux transformation, the higher-order rogue waves on a solitonic background of the KP1 equation are obtained by multiple soliton interactions. In fact, these solitons can evolve to significant and strongly localized transient waves during directional evolution, similarly to KP2 shallow-water interactions. We conjecture that the nth-order rogue wave solution evolves in the form of a triangular extreme wave pattern that consists of n(n+1)2 solitonic lumps in the intermediate time, while only n+1 parallel line solitons possessing equal height are present before and after the initiation of collision dynamics. Such fascinating higher-order rogue waves have not yet been reported in the context of this 2+1 dimensional integrable system. These exact solutions are particularly relevant for the fundamental understanding of extreme wave events in a variety of physical systems, such as plasma and solids.

A new class of nonlinear superposition between lump waves and other waves for Kadomtsev–Petviashvili I equation

In most cases, the lump wave will collide with line waves and breather waves in the hybrid solutions for (2 + 1)-dimensional integrable systems. However, this study introduces a new constraint condition to construct a nonlinear superposition in which the lump wave does not collide with other waves forever. In particular, the soliton molecule consisting of a lump wave, a line wave and any number of breather waves is derived for the first time by the continued introduction of velocity resonance. The new constraints mentioned in this paper will break the traditional understanding of the interaction between a lump and other waves. And this method can be extended to other (2 + 1)-dimensional integrable systems.

Resonance Y-type soliton solutions and some new types of hybrid solutions in the (2+1)-dimensional Sawada–Kotera equation

In this paper, based on N-soliton solutions, we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in (2+1)-dimensional integrable systems. Then, we take the (2+1)-dimensional Sawada–Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint. Next, by the long wave limit method, velocity resonance and module resonance, we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves, breather waves, high-order lump waves respectively. Finally, we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.

Fusion and fission phenomena for [formula omitted]-dimensional fifth-order KdV system

Based on N-soliton solutions, fusion and fission waves are obtained through a new constraint among parameters. With this new constraint, fusion and fission phenomena can be reduced from N-soliton solutions in most (2+1)-dimensional integrable systems. This paper takes the (2+1)-dimensional fifth-order KdV equation as an example to introduce how to use this constraint to generate these fusion and fission waves in detail. At the same time, by continuing to introduce constraints like velocity resonance, module resonance, and long-wave limit, we can generate all kinds of hybrid solutions for granted, such as a hybrid of fissionable waves and breather waves, and a nonlinear superposition between fissionable waves and higher-order lump waves.

Commutator identities and integrable hierarchies

The approach based on commutator identities for elements of associative algebras was previously effectively used to investigate $$(2{+}1)$$ -dimensional integrable systems. We develop this approach to investigate integrable hierarchies and their relations.

Darboux transformation and exact solutions for a (2 + 1)-dimensional integrable system of nonlinear evolution equations

In this paper, starting from a spectral problem, we construct a [Formula: see text]-dimensional integrable system of nonlinear evolution equations. Based on the Lax pair, the recursion operator and Darboux transformation for the whole hierarchy were constructed. As an application, some exact solutions for the hierarchy are obtained by using the Darboux transformation.

Bäcklund transformations and Riemann–Bäcklund method to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, the bilinear method is employed to investigate the N-soliton solutions of a (3 + 1)-dimensional generalized breaking soliton equation. Three sets of bilinear Backlund transformations are obtained by means of gauge transformation. The Riemann–Backlund method is further extended to the (3 + 1)-dimensional nonlinear integrable systems. The quasiperiodic wave solutions of the (3 + 1)-dimensional generalized breaking soliton equation are systematically analyzed. The asymptotic properties of the quasiperiodic solutions are discussed by using a limiting procedure. The one-periodic and two-periodic waves tend to the 1-soliton and 2-soliton under a small amplitude limit, respectively. The dynamical characteristics of the one- and two-periodic waves are summarized by selecting different parameters. Furthermore, we obtain some new types of the quasiperiodic wave solutions of the variable coefficient (3 + 1)-dimensional generalized breaking soliton equation. These solutions present the dynamical behaviors of C-type, anti-C-type and Z-type periodic waves moving on the background of the periodic waves of bell type.

A (1+1)-dimensional integrable system with fifth order spectral problems and four dispersion relations

A new (1+1)-dimensional integrable model is investigated, of which the Lax integrability is shown by the existence of its fifth order spectral problems. Its bilinear form is obtained via introducing an auxiliary variable. The multiple soliton solutions are obtained by solving its bilinear system. It is shown that there are four different dispersion relations for multiple solitons. The amplitudes of the solitons are only wave number dependent while the velocities of the solitons are not only wave number dependent but also dispersion relation dependent. Because of the existence of four dispersion relations, the interactions among solitons are much more abundant because for any fixed wave number there are four different velocities including two left moving and two right moving. Especially, the existence of the multiple velocities makes the velocity resonant conditions can be readily satisfied to form many types of bound states including soliton molecules, soliton-breather molecules and breather molecules.

A differential Galois approach to path integrals

We point out the relevance of the differential Galois theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself [Morales-Ruiz and Ramis, Methods Appl. Anal. 8, 33–96 (2001); see also Morales-Ruiz, in Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Modern Birkhäuser Classics (Springer, Basel, 1999)]; if a finite dimensional complex analytical Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the Galois group of the variational equation around any integral curve must be abelian. A corollary of this result is that, for finite dimensional integrable Hamiltonian systems, the semiclassical approach is computable in a closed form in the framework of the differential Galois theory. This explains in a very precise way the success of quantum semiclassical computations for integrable Hamiltonian systems.

N-soliton and rogue wave solutions of (2+1)-dimensional integrable system with Lax pair

A new generalized (2[Formula: see text]+[Formula: see text]1)-dimensional model, obtained from the Kadomtsev–Petviashvili (KP) equation considered as a system of nonlinear evolution partial differential equations (PDEs), is introduced. With the usage of the Hirota bilinear method and the KP-hierarchy reduction method, N-soliton solutions of the integrable system are constructed. Considering the case of s[Formula: see text]=[Formula: see text]−1 in the linear differential operators, L1 and L2, and a specific set of parameters, two dark solitons, mixed solutions consisting of soliton-type and periodic waves solution are obtained. Based on the particular definition of the matrix elements, one and two rogue waves solutions expressed in terms of rational functions are derived. It is shown that the fundamental rogue waves are line rogue waves, which is different from the property of the moving line solitons of the soliton equations.

The expansions of spectral function and the corresponding finite dimensional integrable systems

In this paper, taking the Lax pairs of KdV equation as an illustrative example, with the help of a transformation of the spectral problem, another form of spectral function is used to give finite dimensional integrable systems. These integrable systems are generated by two different kinds of polynomial expansion of spectral function on spectral parameter. Further, the root variables of the spectral function are introduced to study the canonical equations of Hamilton. Finally, based on the Hamilton-Jacobi theory, the action-angle variables are built and the soliton solutions of KdV equation are obtained by inversion.

DECOMPOSING A NEW NONLINEAR DIFFERENTIAL-DIFFERENCE SYSTEM UNDER A BARGMANN IMPLICIT SYMMETRY CONSTRAINT

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.

Some [formula omitted]-dimensional nonlocal ‘breaking soliton’-type systems

In this paper we propose some (2+1)-dimensional nonlocal ‘breaking soliton’-type systems and show that most of them could be decomposed into two (1+1)-dimensional integrable systems. With the use of this, we construct their Darboux transformations and some exact solutions.

Relations in the cohomology ring of the moduli space of flat SO(2n + 1)-connections on a Riemann surface.

We consider the moduli space of flat SO(2n + 1)-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalizing a conjecture of Newstead.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

The U ( n ) Gelfand–Zeitlin system as a tropical limit of Ginzburg–Weinstein diffeomorphisms

In this paper, we show that the Ginzburg–Weinstein diffeomorphism of Alekseev & Meinrenken (Alekseev, Meinrenken 2007 J. Differential Geom. 76 , 1–34. (10.4310/jdg/1180135664)) admits a scaling tropical limit on an open dense subset of . The target of the limit map is a product , where is the interior of a cone, T is a torus, and carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to recovers the Gelfand–Zeitlin integrable system of Guillemin & Sternberg (Guillemin, Sternberg 1983 J. Funct. Anal. 52 , 106–128. (10.1016/0022-1236(83)90092-7)). As a by-product of our proof, we show that the Lagrangian tori of the Flaschka–Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Open problems, questions and challenges in finite- dimensional integrable systems.

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference 'Finite-dimensional Integrable Systems, FDIS 2017' held at CRM, Barcelona in July 2017.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.