Dromions and a boundary value problem for the Davey-Stewartson 1 equation

doi:10.1016/0167-2789(90)90050-Y

Physica D: Nonlinear Phenomena

Volume 44, Issues 1–2, August 1990, Pages 99-130

https://doi.org/10.1016/0167-2789(90)90050-Y Get rights and content

Abstract

We solve an initial-boundary value problem for the Davey-Stewartson 1 equation, which is a two-dimensional generalization of the nonlinear Schrödinger equation. This equation, which describes the interaction of a surface wave envelope of amplitude q(x, y, t) with the mean flow, arises in a wide range of physical problems. We find that the energy from the mean flow can be transferred to the surface envelope and create focusing effects. Indeed, for generic non-zero boundary conditions on the mean flow, an arbitrary initial q(x, y, t) will form a number of two-dimensional, exponentially decaying in both x and y, localized structures. Furthermore, in contrast to the one-dimensional solitons, these solutions do not preserve their form upon interaction and hence can exchange energy. These coherent structures can be driven everywhere in the plane by choosing a suitable motion for the boundaries. We call these novel localized coherent structures dromions.

References (18)

V.E. Zakharov et al.
Physica D
(1986)
F. Calogero et al.
Inverse Problems
(1987)
F. Calogero et al.
M. Boiti et al.
Phys. Lett. A
(1988)
V.E. Zakharov et al.
Funct. Anal. Appl.
(1974)
D.J. Kaup
J. Math. Phys.
(1981)
A.S. Fokas et al.
Stud. Appl. Math.
(1983)
S.V. Manakov
Physica D
(1981)
A. Davey et al.
D.J. Benney et al.
Stud. Appl. Math.
(1969)
M.J. Ablowitz et al.
Solitons and the Inverse Scattering Transform
(1981)
H. Cornille
J. Math. Phys.
(1979)

There are more references available in the full text version of this article.

Cited by (240)

On the long-time asymptotic for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data
2023, Physica D: Nonlinear Phenomena
From the matrix $\bar{\partial}$ problem, we propose a novel integrable coupled hierarchy with the application of recursive operator $Λ^{n}$ which includes a coupled generalized nonlinear Schrödinger (cgNLS) equations with its Lax pair as $n = 0$ . We successfully derive infinite conservation laws and Hamiltonian function of the cgNLS equations for the first time. Furthermore, we employ the $\bar{\partial}$ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space $H^{1, 1} (R) = H^{1} (R) \cap L^{2, 1} (R)$ . In a fixed space–time cone $S (x_{1}, x_{2}, v_{1}, v_{2}) = {(x, t) \in R^{2} : x = x_{0} + v t, x_{0} \in [x_{1}, x_{2}], v \in [v_{1}, v_{2}]}$ , the long-time asymptotic behavior of the solutions $u (x, t)$ and $v (x, t)$ is derived. Based on the results of asymptotic behavior, we prove the soliton resolution conjecture of the cgNLS equations which consist of three terms, the soliton term confirmed by $| Z (I) |$ -soliton on discrete spectrum, the $t^{- \frac{1}{2}}$ order term on continuous spectrum and the residual error up to $O (t^{- \frac{3}{4}})$ .
Direct separation approach and multi-valued localized excitation for (M+N)-dimensional nonlinear system
2023, Chinese Journal of Physics
In this paper, we propose a new variable separation method that does not need the Hirota’s bilinear form and directly gives the analytic form of the solution $u$ instead of its potential $u_{y}$ . This new method covers the N-soliton solution obtained by the Hirota’s direct method and the multi-valued solution of the multi-linear variable separation approach(MLVSA), and is applicable to the (M+N)-dimensional nonlinear model. We would like to call it the “direct separation approach” (DSA). Taking the extended (3+1)-dimensional Kadomtsev–Petviashvili equation as an example, we introduce an entirely fresh variable separation ansatz and substitute it into the equation, resulting in a new variable separation solution. With the help of this variable separation solution, we construct two types of N-soliton solutions via the single-valued functions. Then, the rogue waves and four typical folded solitary waves are obtained by using the multi-valued functions. In addition, we study the head-on and chase-after collision between two, three, and four folded solitary waves and systematically analyze their dynamic behaviors, such as the phase shifts and their difference values of the interaction. Especially, we detect a multi-directional movement in the collision between four folded solitary waves, which significantly differs from the others. The obtained results may help us simulate complex folded appearance in real life and better understand the wave motion of the solitary waves.
∂̄-dressing method for the (2+1)-dimensional Korteweg–de Vries equation
2023, Applied Mathematics Letters
In this paper, we investigate the (2+1)-dimensional Korteweg–de Vries equation by applying the $\bar{\partial}$ -dressing method. A new $\bar{\partial}$ problem is obtained through the characteristic functions and Green’s functions by Lax representation. A solution of $\bar{\partial}$ problem is constructed by applying Cauchy–Green formula and selecting appropriate spectral transformation. Finally, we can give the solution of the (2+1) dimensional Korteweg–de Vries equation after determining the time evolution of the spectral data.
Mock-integrability and stable solitary vortices
2022, Chaos, Solitons and Fractals
Citation Excerpt :
For that reason, the class of non-linear systems such as KdV and KP is referred to as the completely integrable systems. Alternatively, there have been found various localized solutions to some integrable systems with rapidly decaying in all spatial directions, so-called dromions [7–10], topological multi-vortices [11], localized pulses in regularized-long-wave equations [12], solitons in an extended Burgers equation [13,14], and also several types of lump solitons [15–20], so on. They could explain various nonlinear phenomena of fluid dynamics of the atmosphere or ocean.
Localized soliton-like solutions to a $(2 + 1)$ -dimensional hydro-dynamical evolution equation are studied numerically. The equation is the so-called Williams–Yamagata–Flierl equation, which governs geostrophic fluid in a certain parameter range. Although the equation does not have an integrable structure in the ordinary sense, we find there exist shape-keeping solutions with a very long life in a special background flow and an initial condition. The stability of the localization at the fusion process of two soliton-like objects is also investigated. As for the indicator of the long-term stability of localization, we propose a concept of configurational entropy, which has been introduced in analysis for non-topological solitons in field theories.
The ∂̄-dressing method for the (2+1)-dimensional Konopelchenko–Dubrovsky equation
2022, Applied Mathematics Letters
A novel systematical solution procedure of the (2+1)-dimensional Konopelchenko–Dubrovsky equation is employed on the basis of the $\bar{\partial}$ -dressing method. The eigenfunctions and Green’s function of the Lax pair play a fairly important role in constructing the scattering equation. By analyzing the analyticity of the eigenfunctions and Green’s function, a new $\bar{\partial}$ problem is introduced to help explore the solution with the help of Cauchy–Green formula and choosing the proper spectral transformation. Furthermore, we can work out the solution formally of the Konopelchenko–Dubrovsky equation from inverse spectral problem once the time evolution of the spectral data is determined.
∂¯-dressing method for the nonlocal mKdV equation
2022, Journal of Geometry and Physics
We apply the $\bar{\partial}$ -dressing method to study nonlocal modified Korteweg-de Vries (nonlocal mKdV) equation. A spatial and a time singular spectral problems associated with nonlocal mKdV equation are derived from a local $2 \times 2$ matrix $\bar{\partial}$ -equation via two linear constraint equations. A nonlocal mKdV hierarchy is proposed by using recursive operator. And the conservation laws of the nonlocal mKdV are derived by the temporal linear spectral problem. The N-solitions of the nonlocal mKdV equation are constructed still based the $\bar{\partial}$ -equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained. The results on the mKdV equation can be recovered from the conclusion given above as special reductions.

View all citing articles on Scopus

View full text

Dromions and a boundary value problem for the Davey-Stewartson 1 equation

Abstract

Physica D

Inverse Problems

Phys. Lett. A

Funct. Anal. Appl.

J. Math. Phys.

Stud. Appl. Math.

Physica D

Stud. Appl. Math.

Solitons and the Inverse Scattering Transform

J. Math. Phys.