Intermediate Long Wave Research Articles

Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy

The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer N, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande’s dressing operators for the equivariant Gromov-Witten theory of . This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to 0. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.

Numerical Analysis and Approximate Travelling Wave Solutions for a Higher Order Internal Wave System

In this work we focus on the numerical solution of a higher order bidirectional nonlinear model of Boussinesq type involving a nonlocal operator. Based on a von Neumann stability analysis for the linearized problem, an efficient and stable scheme for the nonlinear system is proposed. Our method is based on a numerical scheme known from the literature that solves satisfactorily a lower order linear system. Additionally, approximate periodic travelling wave solutions profiles for the higher order nonlinear system are presented. Such approximate travelling wave solutions are obtained from a solitary wave family of solutions for the Intermediate Long Wave (ILW) equation and the regularized Intermediate Long Wave (rILW) equation.

Bäcklund transformation, infinite number of conservation laws and fission properties of an integro-differential model for ocean internal solitary waves

This paper presents an analytical investigation of the propagation of internal solitary waves in the ocean of finite depth. Using the multi-scale analysis and reduced perturbation methods, the integro-differential equation is derived, which is called the intermediate long wave (ILW) equation and can describe the amplitude of internal solitary waves. It can reduce to the Benjamin–Ono equation in the deep-water limit, and to the KdV equation in the shallow-water limit. Little attention has been paid to the features of integro-differential equations, especially for their conservation laws. Here, based on Hirota bilinear method, Bäcklund transformations in bilinear form of ILW equation are derived and infinite number of conservation laws are given. Finally, we analyze the fission phenomenon of internal solitary waves theoretically and verify it through numerical simulation. All of these have potential value for the further research on ocean internal solitary waves.

On the Long Time Behavior of Solutions to the Intermediate Long Wave Equation

We show that the limit infimum, as time $\,t\,$ goes to infinity, of any uniformly bounded in time $H^{3/2+}\cap L^1$ solution to the intermediate long wave (ILW) equation converges to zero locally...

Nonchiral intermediate long-wave equation and interedge effects in narrow quantum Hall systems

We present a non-chiral version of the Intermediate Long Wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account inter-edge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws for this new equation. We also present a periodic non-chiral ILW equation, together with its soliton solutions governed by the elliptic CMS model.

Boundary layer collapses described by the two-dimensional intermediate long-wave equation

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter D. In the limits of large and small D, the 2d-ILW equation respectively tends to the 2d Benjamin-Ono and 2d Zakharov-Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude a and width σ, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane (a, σ) for various values of D. The amplitude threshold a increases as D and σ decrease and tends to infinity at D → 0. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all D except D = 0. But the equation itself has not been proved for small D. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.

Weakly and strongly coupled intermediate long-wave hierarchies and Benjamin–Ono equations

In this paper, we construct weakly and strongly coupled intermediate long-wave (ILW) hierarchies which contain integrable weakly (strongly) coupled Benjamin–Ono2 equations as the primary equations. The dressing operators [Formula: see text] of the weakly and strongly coupled ILW hierarchies satisfy the Sato equations of a weakly (strongly) coupled KP hierarchy. We found an infinite system of commuting integrals of motion related to the weakly and strongly coupled Benjamin–Ono integrable hierarchies.

Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics

We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on C^2 x S^2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S^2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W_N algebrae, thus providing a gauge theoretical proof of AGT correspondence.

Intermediate long wave systems for internal waves

This paper, which deals with internal waves in the two-layer formulation, consists of three parts. The first part is devoted to the derivation of asymptotic models in the intermediate long wave (ILW) regime for internal waves with a free upper surface and with a flat bottom. Using a method similar to that introduced by Bona et al (2008 J. Math. Pures Appl. 89 538–66), we obtain a one-parameter ILW system which is consistent with the full Euler system. In the second part we investigate the well-posedness of the ILW system under the rigid lid assumption. The local well-posedness on the time scale in lower order Sobolev spaces and the large time well-posedness on the long time scale in higher order Sobolev spaces are proven for both 1D and 2D cases. The long time existence result seems to be the first of this type in the context of internal waves. In the third part we provide a rigorous justification to show that the Benjamin–Ono (BO) system can be deduced from the ILW system by letting the depth of the lower flow tend to ∞. The convergence results between the solutions of the BO system and the ILW system are established in both the short time scale and the long time scale , in different functional settings.

Global existence and asymptotic behavior of solutions to the homogeneous Neumann problem for ILW equation on a half-line

We consider the homogeneous Neumann initial-boundary value problem for intermediate long-wave (ILW) equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

Two-dimensional interactions of solitons in a two-layer fluid of finite depth

Two-dimensional (2D) interactions of two interfacial solitons in a two-layer fluid of finite depth are investigated under the assumption of a small but finite amplitude. When the angle ϑ between the wave normals of two solitons is not small, it is shown by a perturbation method that in the lowest order of approximation the solution is a superposition of two intermediate long wave (ILW) solitons and in the next order of approximation the effect of the interaction appears as position phase shifts and as an increase in amplitude at the interaction center of two solitons. When ϑ is small, it is shown that the interaction is described approximately by a nonlinear integro-partial differential equation that we call the two-dimensional ILW (2DILW) equation. By solving it numerically for a V-shaped initial wave that is an appropriate initial value for the oblique reflection of a soliton due to a rigid wall, it is shown that for a relatively large angle of incidence ϑi the reflection is regular, but for a relatively small ϑi the reflection is not regular and a new wave called stem is generated. The results are also compared with those of the Kadomtsev–Petviashvili (KP) equation and of the two-dimensional Benjamin–Ono (2DBO) equation.

Periodic ILW equation with discrete Laplacian

We study an integro-differential equation which generalizes the periodic intermediate long wave (ILW) equation. The kernel of the singular integral involved is an elliptic function written as a second-order difference of the Weierstrass ζ-function. Using Sato's formulation, we show the integrability and construct some special solutions. An elliptic solution is also obtained. We present a conjecture based on a Poisson structure that gives an alternative description of this integrable hierarchy. We note that this Poisson algebra in turn is related to a quantum algebra related to the family of Macdonald difference operators.

A Higher‐Order Internal Wave Model Accounting for Large Bathymetric Variations

A higher‐order strongly nonlinear model is derived to describe the evolution of large amplitude internal waves over arbitrary bathymetric variations in a two‐layer system where the upper layer is shallow while the lower layer is comparable to the characteristic wavelength. The new system of nonlinear evolution equations with variable coefficients is a generalization of the deep configuration model proposed by Choi and Camassa [1] and accounts for both a higher‐order approximation to pressure coupling between the two layers and the effects of rapidly varying bottom variation. Motivated by the work of Rosales and Papanicolaou [2], an averaging technique is applied to the system for weakly nonlinear long internal waves propagating over periodic bottom topography. It is shown that the system reduces to an effective Intermediate Long Wave (ILW) equation, in contrast to the Korteweg‐de Vries (KdV) equation derived for the surface wave case.

A reduced model for internal waves interacting with topography at intermediate depth

A reduced strongly nonlinear model is derived for the evolution of one-dimensional internal waves over an arbitrary bottom topography. Two layers containing inviscid, immiscible, irrotational fluids of different densities are defined. The upper layer is shallow compared with the characteristic wavelength at the interface, while the bottom region’s depth is comparable to the wavelength. The nonlinear evolution equations obtained are in terms of the internal wave elevation and the mean upper-velocity for the configuration described. The system is a generalization of the one proposed by Choi and Camassa for the flat bottom case in the same physical settings. Due to the presence of a topography a variable coefficient system of partial differential equations arises. These Boussinesq-type equations contain the Intermediate Long Wave (ILW) equation and the Benjamin- Ono (BO) equation when restricted to the unidirectional propagation regime. We intend to use this model to study the interaction of waves with the bottom profile. The dynamics include wave scattering, dispersion and attenuation, among other phenomena.

An application of the inverse scattering transform to the modified intermediate long wave equation

The modified intermediate long wave (MILW) equation is a (1+1)-dimensional nonlinear singular integro-differential equation that possesses soliton solutions. In an appropriate limit the MILW equation reduces to the well-known modified Korteweg-de Vries equation. In this paper we solve the initial value problem for the MILW equation through a suitable implementation of the inverse scattering transform and use of the Miura-type transformation that maps solutions of the MILW equation into solutions of a complexified version of the standard intermediate long wave (ILW) equation. The initial value used for the MILW equation is assumed to be real valued, sufficiently smooth, and decaying to zero as the absolute value of the spatial variable approaches large values. An interesting feature of the procedure we develop is that soliton solutions for the ILW and MILW equations can be derived by appropriate specializations of a master set of equations.

On the ILW hierarchy

In this Letter, we present a new hierarchy which includes the intermediate long wave (ILW) equation at the lowest order. This hierarchy is thought to be a novel reduction of the 1st modified KP type hierarchy. The framework of our investigation is Sato theory.

Fully nonlinear internal waves in a two-fluid system

Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

On the number of solitons for the intermediate long wave equation

In this work we use an extension of the WKB method to nonlocal problems to count number of solitons produced by an arbitrary slowly varying initial disturbance in the ILW (Intermediate Long Wave) evolution equation. The results give the complete description for the range of parameters. In the course of the analysis we also recover the results for the Benjamin-Ono equation, recently obtained by dimensional arguments and verified by numerical simulations by Miloh et al. [5] and ingeniously by Matsuno [3], under a specific and still unproved assumption regarding the additive behavior of the conservation laws for the Benjamin-Ono equation.

Oblique Interactions between Internal Solitary Waves

In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ1=|cm/cn−cosδ|, Δ2=|cn/cm−cosδ|,cm,n, are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg‐de Vries (KdV) equation for a shallow fluid, or intermediate long‐wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ1,2 are 0(α), and this case is governed by two coupled Kadomtsev‐Petviashvili (KP) equations for a shallow fluid, or two coupled two‐dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ1 is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ1 leads to a local distortion of the wave speed of the mode n. When the interacting waves belong to the same mode (i.e., m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.

Periodic solutions of the intermediate long-wave equation: a nonlinear superposition principle

Periodic stationary-wave solutions of the intermediate long-wave (ILW) equation are derived using the bilinear transformation method, and a new expression for the dispersion relation is obtained. The class of physically important real-valued solutions is identified. These solutions may be represented as an infinite superposition of solitary-wave profiles, a property shared by the related Korteweg-de Vries (KdV) and Benjamin-Ono (BO) equation. This nonlinear superposition principle, which has been the subject of various interpretations in the literature, is discussed. The ILW periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes, respectively. For intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave. In the shallow-water (KdV) limit the ILW periodic solution leads to the familiar cnoidal wave, whereas the deep-water (BO) limit yields Benjamin's periodic wave. A previously unknown expression for the cnoidal-wave dispersion relation in terms of theta functions is obtained.