Dispersionless Limit Research Articles

Propagation of wave packets along large-scale background waves

We study propagation of high-frequency wave packets along a large-scale background wave, which evolves according to dispersionless hydrodynamic equations for two variables (fluid density and flow velocity). Influence of the wave packet on evolution of the background wave is neglected, so the large-scale evolution can be found independently of the wave packet's motion. At the same time, propagation of the packet depends in an essential way on the background wave, and it can be considered in a framework of the geometric optics approximation with the use of Hamilton equations for the carrier wave number and the mean co-ordinate of the packet. We derive equations for the carrier wave number as a function of the parameters, which describe the background wave. When they are solved, the path of the packet can be found by simple integration of the Hamilton equation. The theory is illustrated by its application to the problem of propagation of wave packets along expanding a large-scale wave, in which evolution is described by the shallow water equations. In particular, they correspond to the dispersionless limit of the defocusing nonlinear Schrödinger equation, and then the expanding wave can be considered as an expanding cloud of the Bose–Einstein condensate. Reflection of wave packets from upstream flows and their propagation along stationary flows are also discussed. The analytical solutions found for these particular cases agree very well with an exact numerical solution of the nonlinear Schrödinger equation.

Low-dimensional bihamiltonian structures of topological type

We construct local bihamiltonian structures from classical W-algebras associated with non-regular nilpotent elements of regular semisimple type in Lie algebras of types A2 and A3. They form exact Poisson pencils and admit a dispersionless limit, and their leading terms define logarithmic or trivial Dubrovin–Frobenius manifolds. We calculate the corresponding central invariants, which are expected to be constants. In particular, we get Dubrovin–Frobenius manifolds associated with the focused Schrödinger equation and Hurwitz space M0;1,0 and the corresponding bihamiltonian structures of topological type.

Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation

We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov-Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they are characterized by a differential equation of the L\"owner type which we call the symmetric radial L\"owner equation. It is also shown that solutions to the symmetric radial L\"owner equation are conformal maps of the exterior of the unit circle with two symmetric slits to the exterior of the unit circle.

A GENERALISATION OF WITTEN’S CONJECTURE FOR THE PIXTON CLASS AND THE NONCOMMUTATIVE KDV HIERARCHY

AbstractIn this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.

Nonlinear Transformation of Sine Wave within the Framework of Symmetric (2+4) KdV Equation

This paper considers the transformation of a sine wave in the framework of the extended modified Korteweg–de Vries equation or (2+4) KdV, which includes a combination of cubic and quintic nonlinearities. It describes the internal waves in a medium with symmetric vertical density stratification, and all the considerations in this study are produced for the reasonable combinations of the signs of the coefficients for nonlinear and dispersive terms, provided by this physical problem. The features of Riemann waves—solutions of the dispersionless limit of the model—are described in detail: The times and levels of breaking are derived in an implicit analytic form depending on the amplitude of the initial sine wave. It is demonstrated that the shock occurs at two (for small amplitudes) or four (for moderate and large amplitudes) levels per period of sine wave. Breaking at different levels occurs at different times. The symmetric (2+4) KdV equation is not integrable, but nevertheless it has stationary solutions in the form of traveling solitary waves of both polarities with a limiting amplitude. With the help of numerical calculations, the features of the processes of a sinusoidal wave evolution and formation of undular bores are demonstrated and analyzed. Qualitative features of multiple inelastic interactions of emerging soliton-like pulses are displayed.

Tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy

We construct a local tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy, and compute the central invariants of the associated bihamiltonian structures. We show that the central invariants of one of the bihamiltonian structures are equal to 124, and the dispersionless limit of this bihamiltonian structure coincides with the one that is defined on the jet space of the Frobenius manifold associated with the Gromov–Witten invariants of local P1. This result provides support for the validity of Brini’s conjecture on the relation of these Gromov–Witten invariants with the Ablowitz–Ladik hierarchy.

Kadomtsev–Petviashvili hierarchies of types B and C

This is a short review of the Kadomtsev–Petviashvili hierarchies of types $$\mathrm{B}$$ and $$\mathrm{C}$$ . The main objects are the $$L$$ operator, the wave operator, the auxiliary linear problems for the wave function, the bilinear identity for the wave function, and the tau function. All of them are discussed in the paper. Connections with the usual (type- $$\mathrm{A}$$ ) Kadomtsev–Petviashvili hierarchy are clarified. Examples of soliton solutions and the dispersionless limit of the hierarchies are also considered.

Hamiltonian Aspects of Three-Layer Stratified Fluids

The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

КВАЗИКЛАССИЧЕСКИЙ ПРЕДЕЛ УРАВНЕНИЙ ШРЕДИНГЕРА-МАКСВЕЛЛА-БЛОХА

The study of integrable equations is one of the most important aspects of modern mathematical and theoretical physics. Currently, there are a large number of nonlinear integrable equations that have a physical application. The concept of nonlinear integrable equations is closely related to solitons. An object being in a nonlinear medium that maintains its shape at moving, as well as when interacting with its own kind, is called a soliton or a solitary wave. In many physical processes, nonlinearity is closely related to the concept of dispersion. Soliton solutions have dispersionless properties. Connection with the fact that the nonlinear component of the equation compensates for the dispersion term. In addition to integrable nonlinear differential equations, there is also an important class of integrable partial differential equations (PDEs), so-called the integrable equations of hydrodynamic type or dispersionless (quasiclassical) equations [1-13]. Nonlinear dispersionless equations arise as a dispersionless (quasiclassical) limit of known integrable equations. In recent years, the study of dispersionless systems has become of great importance, since they arise as a result of the analysis of various problems, such as physics, mathematics, and applied mathematics, from the theory of quantum fields and strings to the theory of conformal mappings on the complex plane. Well-known classical methods of the theory of intrinsic systems are used to study dispersionless equations. In this paper, we present the quasicalassical limit of the system of (1+1)-dimensional Schrödinger-Maxwell- Bloch (NLS-MB) equations. The system of the NLS-MB equations is one of the classic examples of the theory of nonlinear integrable equations. The NLS-MB equations describe the propagation of optical solitons in fibers with resonance and doped with erbium. And we will also show the integrability of the quasiclassical limit of the NLS-MB using the obtained Lax representation.

Plasma slab expansion into vacuum

The problem of collisionless plasma slab expansion into vacuum is solved within a two-temperature hydrodynamic approximation in the dispersionless limit of zero Debye radius. In the framework of such an approach, the solution by the Riemann method provides a quite accurate description of the whole process of plasma dynamics. It is shown that the dispersionless approximation agrees very well with exact numerical solution of the full system of plasma hydrodynamic equations.

On a class of 2D integrable lattice equations

We develop a new approach to the classification of integrable equations of the form uxy=f(u,ux,uy,△zu△z¯u,△zz¯u), where △z and △z¯ are the forward/backward discrete derivatives. The following two-step classification procedure is proposed: (1) First, we require that the dispersionless limit of the equation is integrable, that is, its characteristic variety defines a conformal structure, which is Einstein–Weyl, on every solution. (2) Second, to the candidate equations selected at the previous step, we apply the test of Darboux integrability of reductions obtained by imposing suitable cutoff conditions.

Dispersionless limit of the (1+1)-dimensional Fokas-Lenells equation

Nonlinear dispersionless equations can be obtained as dispersionless limits (quasiclassical limits) of integrable hierarchies of equations or by constructing a system of hydrodynamic type. In this paper, the dispersionless limit of the (1+1)-dimensional Fokas-Lenells equation is found. We present Lax pair for this dispersionless equation. As well known, the Fokas-Lenells equation characterizes the propagation of ultrashort nonlinear light pulses in optical fibers. Exact solutions of the dispersionless version of the Fokas-Lenells equation describe nonlinear shock waves in optical fiber systems. By using the obtained results, one can find shock wave solutions of the dispersionless equation, which have different physical applications.

Multi‐dimensional conservation laws and integrable systems

AbstractIn this paper, we introduce a new property of two‐dimensional integrable hydrodynamic chains—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Infinitely many local three‐dimensional conservation laws for the Benney commuting hydrodynamic chains are constructed. As a by‐product, we established a new method for computation of local conservation laws for three‐dimensional integrable systems. The Mikhalëv equation and the dispersionless limit of the Kadomtsev‐Petviashvili equation are investigated. All known local and infinitely many new quasilocal three‐dimensional conservation laws are presented. Also four‐dimensional conservation laws are considered for couples of three‐dimensional integrable quasilinear systems and for triplets of corresponding hydrodynamic chains.

Dispersionless Limits of Ma Equations

Dispersionless Limits of Ma Equations

Integrability of type 0A matrix model in the presence of D-brane

We consider type 0A matrix model in the presence of spacelike D-brane which is localized in matter direction at any arbitrary point. In string theory, the boundary state, which in matrix model corresponds to the Laplace transform of the macroscopic loop operator, is known to obey the operator constraints corresponding to open string boundary condition. When we analyze MQM as well as the respective collective field theory and compare it with dual string theory, it appears that consistency of the theory requires a condition equivalent to a constraint on the matter part that needed to be imposed in the matrix model. We identified this condition and observed that this results in constraining the macroscopic loop operator so that it projects the Hilbert space generated by the operator to its physical sector at the point of insertion while keeping the bulk matrix model unaffected, thereby describing a situation parallel to string theory. We analyzed the theory with uncompactified time and have shown explicitly that the matrix model predictions are in good agreement with the relevant string theory. Next, we considered the theory with compactified time, analyzed MQM on a circle in the presence of D-brane. We evaluated the partition function along with the constrained macroscopic loop operator in the grand canonical ensemble and showed the free energy corresponds to that of a deformed Fermi surface. We have also shown that the path integral in the presence of D-brane can be expressed as the Fredholm determinant. We have studied the fermionic scattering in a semiclassical regime. Finally, we considered the compactified theory in the presence of the D-brane with tachyonic background. We evaluated the free energy in the grand canonical ensemble. We have shown the integrable structure of the respective partition function and it corresponds to the tau function of Toda hierarchy. We have also analyzed the dispersionless limit.

Dispersionless limit of the Konno-Oono equation

Dispersionless limit of the Konno-Oono equation

The WDVV Associativity Equations as a High-Frequency Limit

In this paper, we present a new “Hamiltonian” approach for construction of integrable systems. We found an intermediate dispersive system of a Camassa–Holm type. This three-component system has simultaneously a high-frequency (short wave) limit equivalent to the remarkable WDVV associativity equations and a dispersionless (long wave) limit coinciding with a dispersionless limit of the Yajima–Oikawa system.

Flat F-manifolds, Miura invariants, and integrable systems of conservation laws

We extend some of the results proved for scalar equations in [3,4], to the case of systems of integrable conservation laws. In particular, for such systems we prove that the eigenvalues of a matrix obtained from the quasilinear part of the system are invariants under Miura transformations and we show how these invariants are related to dispersion relations. Furthermore, focusing on one-parameter families of dispersionless systems of integrable conservation laws associated to the Coxeter groups of rank $2$ found in [1], we study the corresponding integrable deformations up to order $2$ in the deformation parameter $\epsilon$. Each family contains both bi-Hamiltonian and non-Hamiltonian systems of conservation laws and therefore we use it to probe to which extent the properties of the dispersionless limit impact the nature and the existence of integrable deformations. It turns out that a part two values of the parameter all deformations of order one in $\epsilon$ are Miura-trivial, while all those of order two in $\epsilon$ are essentially parameterized by two arbitrary functions of single variables (the Riemann invariants) both in the bi-Hamiltonian and in the non-Hamiltonian case. In the two remaining cases, due to the existence of non-trivial first order deformations, there is an additional functional parameter.

Shock wave occurrence and soliton propagation in polariton condensates

We study the effects of the gain and the loss of polaritons on the wave propagation in polariton condensates. This system is described by a modified Gross–Pitaevskii equation. In the case of small damping, we use the reductive perturbation method to transform this equation; we get a modified Burgers equation in the dispersionless limit and a damped Korteweg – de Vries equation in a more general case. We demonstrate that the shock wave occurrence depends on the gain and the loss of polaritons in the dispersionless polariton condensate. The resolution of the damped Korteweg – de Vries equation shows that the soliton behaves like a damped wave in the case of a constant damping. Based on an asymptotic solution, the survival time and the distance traveled by this soliton are evaluated. We solve the damped Korteweg – de Vries equation and the modified Gross–Pitaevskii numerically to validate the analytical calculations and discuss especially the soliton propagation in the system.

Multi-variable reductions of the dispersionless DKP hierarchy

We consider multi-variable reductions of the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) in the elliptic parametrization. The reduction is given by a system of elliptic Löwner equations supplemented by a system of partial differential equations of hydrodynamic type. The compatibility conditions for the elliptic Löwner equations are derived. They are elliptic analogues of the Gibbons–Tsarev equations. We prove solvability of the hydrodynamic type system by means of the generalized hodograph method. The associated diagonal metric is proved to be of the Egorov type.