Intermediate Long Wave Equation Research Articles

Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Bäcklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin–Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

Spin generalizations of the Benjamin–Ono equation

We present new soliton equations related to the A-type spin Calogero–Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin–Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO 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.

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.

Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations

Considered here are systems of partial differential equations arising in internal wave theory. The systems are asymptotic models describing the two-way propagation of long-crested interfacial waves in the Benjamin-Ono and the Intermediate Long-Wave regimes. Of particular interest will be solitary-wave solutions of these systems. Several methods of numerically approximating these solitary waves are put forward and their performance compared. The output of these schemes is then used to better understand some of the fundamental properties of these solitary waves. The spatial structure of the systems of equations is non-local, like that of their one-dimensional, unidirectional relatives, the Benjamin-Ono and the Intermediate Long-Wave equations. As the non-local aspect is comprised of Fourier multiplier operators, this suggests the use of spectral methods for the discretization in space. Three iterative methods are proposed and implemented for approximating traveling-wave solutions. In addition to Newton-type and Petviashvili iterations, an interesting wrinkle on the usual Petviashvili method is put forward which appears to offer advantages over the other two techniques. The performance of these methods is checked in several ways, including using the approximations they generate as initial data in time-dependent codes for obtaining solutions of the Cauchy problem. Attention is then turned to determining speed versus amplitude relations of these families of waves and their dependence upon parameters in the models. There are also provided comparisons between the unidirectional and bidirectional solitary waves. It deserves remark that while small-amplitude solitary-wave solutions of these systems are known to exist, our results suggest the amplitude restriction in the theory is artificial.

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.

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

В рамках существенно двумерного обобщения уравнения Джозефа исследуется нелинейная динамика локализованных возмущений типичного сдвигового течения пограничного слоя в жидкости между двумя параллельными плоскостями, расстояние между которыми характеризуется безразмерным параметром $D$. В пределе больших и малых $D$ это уравнение переходит соответственно в двумерные уравнения Бенджамина-Оно и Захарова-Кузнецова. Показано, что если гамильтониан отрицателен, то любые локализованные начальные возмущения коллапсируют, т. е. взрывным образом растут, за конечное время образуя точечную сингулярность. Это происходит, когда амплитуда начального возмущения превышает определенный порог, зависящий от вида начального возмущения. Для аксиально-симметричных гауссовых и лоренцевых начальных возмущений с амплитудой $a$ и шириной $\sigma$ при различных значениях $D$ в явном виде получены нелинейные нейтральные кривые на плоскости $(a,\sigma)$, разделяющие области коллапса и затухания возмущения. Пороговое значение амплитуды $a$ растет с уменьшением $D$ и $\sigma$ и стремится к бесконечности при $D\to 0$. Двумерное уравнение Джозефа допускает стационарные аксиально-симметричные солитонные решения, для которых гамильтониан всегда отрицателен, они коллапсируют для всех $D$, кроме $D=0$. Однако для малых $D$ не доказана справедливость самого́ уравнения. С помощью прямого численного моделирования с гауссовыми и лоренцевыми начальными условиями показано, что начальные возмущения с амплитудой, превышающей найденный порог, коллапсируют самоподобным образом, а возмущения с амплитудами ниже порога затухают.

Self-similar asymptotics for solutions to the intermediate long-wave equation

We consider the Cauchy problem for the intermediate long-wave equation $$\begin{aligned} u_{t}-\partial _{x}u^{2}+\frac{1}{\vartheta }u_{x}+VP\int _{\mathbb {R}}\frac{1}{2\vartheta }\coth \left( \frac{\pi \left( y-x\right) }{2\vartheta }\right) u_{yy}\left( t,y\right) \mathrm{d}y=0, \end{aligned}$$ where $$\vartheta >0$$ . Our purpose in this paper is to prove the large time asymptotic behavior of solutions under the nonzero mass condition $$\int u_{0}\left( x\right) \mathrm{d}x\ne 0$$ .

On the modified intermediate long-wave equation

We consider the modified intermediate long-wave equationWe develop the factorization technique to study the large time asymptotics of solutions.

On genus-two solutions for the ILW equation

The existence of theta function solutions of genus two for the intermediate long-wave equation is established. A numerical example is also presented. The method basically goes along with Krichever’s construction of theta function solutions for soliton equations, such as the Kronecker product equation. This idea leads us to a question whether a Riemann surface exists which allows a peculiar abelian integral of the third kind. The answer is affirmative at least for genus-two curves.

Well-posedness for the regularized intermediate long-wave equation

Well-posedness for the regularized intermediate long-wave equation

Five-wave classical scattering matrix and integrable equations

We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.

Mixed initial-boundary value problem for intermediate long-wave equation

We consider the mixed initial-boundary value problem for intermediate long-wave equation on the half-line. We study the several rigorous aspects of this problem including global in time existence of solutions and the asymptotic behavior of solutions for large time. The type of existence results derived here plays a crucial role in the rigorous investigation of integrable equations.

Dirichlet problem for intermediate long-wave equation

We consider the initial-boundary-value problem for the 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.

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.

Intermediate long-wave equation on a half-line

We consider the initial-boundary value problem for intermediate long-wave 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.

On the uniqueness of ground states of non-local equations

We review our joint result with E. Lenzmann about the uniqueness of ground state solutions of non-linear equations involving the fractional Laplacian and provide an alternate uniqueness proof for an equation related to the intermediate long-wave equation.

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.

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.