System Of Nonlinear Evolution Equations Research Articles

Extended rotation and scaling groups for nonlinear evolution equations

A (1+1)-dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator V = x∂ u − u∂ x . Then the solution satisfies the condition u x = −x/u. For equations that do not admit the rotation group, we provide an extension of the rotation group. The corresponding exact solution can be constructed via the invariant set R 0 = {u: u x = xF(u)} of a contact first-order differential structure, where F is a smooth function to be determined. The time evolution on R 0 is shown to be governed by a first-order dynamical system. We introduce an extension of the scaling groups characterized by an invariant set $$\tilde S_0 $$ that depends on two constants ∈ and n ≠ 1. When ∈ = 0, it reduces to the invariant set S 0 introduced by Galaktionov. We also introduce a generalization of both the scaling and rotation groups, which is described by an invariant set E 0 with parameters a and b. When a = 0 or b = 0, it respectively reduces to R 0 or S 0. These approaches are used to obtain exact solutions and reductions of dynamical systems of nonlinear evolution equations.

Extended rotation and scaling groups for nonlinear evolution equations

A (1+1)-dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator V = x∂ u − u∂ x . Then the solution satisfies the condition ux = −x/u. For equations that do not admit the rotation group, we provide an extension of the rotation group. The corresponding exact solution can be constructed via the invariant set R0 = {u: ux = xF(u)} of a contact first-order differential structure, where F is a smooth function to be determined. The time evolution on R0 is shown to be governed by a first-order dynamical system. We introduce an extension of the scaling groups characterized by an invariant set \(\tilde S_0 \) that depends on two constants ∈ and n ≠ 1. When ∈ = 0, it reduces to the invariant set S0 introduced by Galaktionov. We also introduce a generalization of both the scaling and rotation groups, which is described by an invariant set E0 with parameters a and b. When a = 0 or b = 0, it respectively reduces to R0 or S0. These approaches are used to obtain exact solutions and reductions of dynamical systems of nonlinear evolution equations.

Interacting dromions for electron acoustic waves

Analytical and numerical investigation of electron acoustic waves shows the existence of interacting dromions solutions. Using the asymptotic perturbation (AP) method, based on Fourier expansion and spatio-temporal rescaling, it is found that the amplitude slow modulation of Fourier modes is described by a system of nonlinear evolution equations. This system is C-integrable, i.e. can be linearized through an appropriate transformation of the dependent and independent variables. We demonstrate the existence of dromion solutions, which propagate with their own group velocity and during a collision maintain their shape, the only change being a phase shift. Numerical results are used to check the validity of the AP method.

Navier-Stokes equations on Lipschitz domains in Riemannian manifolds

The Navier-Stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous, incompressible fluid. One ingredient in the analysis of this system is the stationary, linear system known as the Stokes system, a boundary value problem (BVP) that will be described in detail in the next section. Layer potential methods in smoothly bounded domains in Euclidean space have proven useful in the analysis of the Stokes system, starting with work of Odqvist and Lichtenstein, and including work of Solonnikov and many others. See the discussion in Chapter III of [10] and in [17], for the case of flow in regions with smooth boundary. A treatment based on the modern language of pseudodifferential operators can be found in [18]. In 1988, E. Fabes, C. Kenig and G. Verchota [6], extended this classical layer potential approach to cover Lipschitz domains in Euclidean space. In [6] the main result concerning the (constant coefficient) Stokes system on Lipschitz domains with connected boundary in Euclidean space, is the treatment of the L-Dirichlet boundary value problem (and its regular version). To achieve this, the authors solve certain auxiliary Neumann type problems and then exploit the duality between these and the original BVP’s at the level of boundary integral operators. P. Deuring and W. von Wahl [4] made use of the analysis in [6] to demonstrate the short-time existence of solutions to the Navier-Stokes equations in bounded Lipschitz domains in threedimensional Euclidean space. It was necessary in [4] to include the hypothesis that the boundary be connected. The hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on Lipschitz domains. It was certainly natural to speculate that this restriction was an artifact of the methods used and not ∗Partly supported by NSF grant DMS-9870018. †Partially supported by NSF grant DMS-9877077. 1991Mathematics Subject Classification. Primary 35Q30, 76D05, 35J25; Secondary 42B20, 45E05.

Non-resonant interacting water waves in 2+1 dimensions

We investigate the interaction among small amplitude water waves, when the fluid motion is in a basin of arbitrary, uniform depth. Waves are supposed to be non-resonant, i.e., with different group velocities that are not close to each other. Starting from the isotropic pseudo-differential Milewski–Keller equation and using an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling, we show that the amplitude slow modulation of Fourier modes can be described by a model system of non-linear evolution equations. We demonstrate that the system is C-integrable, i.e., can be linearized through an appropriate transformation of the dependent and independent variables. A subclass of solutions gives rise to non-localized line-solitons and localized solitons (dromions). Each soliton propagates with the group velocity and during a collision maintains its shape, the only change being a phase shift.

Study on exact analytical solutions for two systems of nonlinear evolution equations

The homogeneous balance method was improved and applied to two systems of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.

The moser type reduction of integrable Riccati differential equations and its Lie-algebraic structure

A given Riccati equation, as is well known, can be naturally reduced to a system of nonlinear evolution equations on an infinite-dimensional functional manifold with Cauchy-Goursat initial data. We describe the Lie algebraic reduction procedure of nonlocal type for this infinite-dimensional dynamical system upon the set of critical points of an invariant Lagrangian functional. As one of our main results, we show that the reduced dynamical system generates the completely integrable Hamiltonian flow on this submanifold with respect to the canonical symplectic structure upon it. The above also makes it possible to find effectively its finite-dimensional Lax type representation via both the well known Moser type reduction procedure and the dual momentum mapping scheme on some matrix manifold.

On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics

According to the improved sine-cosine method and Wu-elimination method, a new algorithm to construct solitary wave solutions for systems of nonlinear evolution equations is put forward. The algorithm has some conclusions which are better than what the hyperbolic function method known does and simpler in use. With the aid of MATHEMATICA, the algorithm can be carried out in computer.

Geometric description of a relativistic string

Classical three-dimensional relativistic string theory is considered in terms of world sheet quadratic forms. Taking the second quadratic form, not only the first one, into account is essential. A system of nonlinear evolution equations describing the string dynamics at the surface of primary constraints in a conformally invariant manner is derived. The results are generalized to the four-dimensional case.

Quadratic solitons in nonlinear dynamical lattices

We propose a lattice model, in both one- and multidimensional versions, which may give rise to matching conditions necessary for the generation of solitons through the second-harmonic generation. The model describes an array of linearly coupled two-component dipoles in an anisotropic nonlinear host medium. Unlike this discrete system, its continuum counterpart gives rise to the matching conditions only in a trivial degenerate situation. A system of nonlinear evolution equations for slowly varying envelope functions of the resonantly coupled fundamental- and second-harmonic waves is derived. In the one-dimensional case, it coincides with the standard system known in nonlinear optics, which gives rise to stable solitons. In the multidimensional case, the system proves to be more general than its counterpart in optics, because of the anisotropy of the underlying lattice model.

Nonresonant interacting waves for the nonlinear Klein–Gordon equation in three-dimensional space

Interaction among nonresonant waves of the nonlinear Klein–Gordon equation in ordinary (three-dimensional) space is investigated, by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. We show that the slow amplitude modulation of Fourier modes can be described by a system of nonlinear evolution equations. The system is C-integrable, i.e. can be linearized through an appropriate transformation of the dependent variables. N-period quasiperiodic solutions with a nonlinear dispersion relation are observed. Moreover, envelope solitons with fixed but arbitrary shapes and velocities connected to the group velocities of the carrier waves are possible. During a collision, solitons maintain their shape, but are subjected to a phase shift. The technique proposed in this paper can be applied to the description of soliton interactions in nonlinear dispersive media without using the complexity of the inverse scattering method.

Non-resonant interacting ion acoustic waves in a magnetized plasma

We perform an analytical and numerical investigation of the interaction among non-resonant ion acoustic waves in a magnetized plasma. Waves are supposed to be non-resonant, i.e. with different group velocities that are not close to each other. We use an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. We show that the amplitude slow modulation of Fourier modes cannot be described by the usual nonlinear Schrodinger equation but by a new model system of nonlinear evolution equations. This system is C-integrable, i.e. it can be linearized through an appropriate transformation of the dependent and independent variables. We demonstrate that a subclass of solutions gives rise to envelope solitons. Each envelope soliton propagates with its own group velocity. During a collision solitons maintain their shape, the only change being a phase shift. Numerical results are used to check the validity of the asymptotic perturbation method.

Role of parabolic viscosity in numerical analysis of derivative nonlinear evolution equations

We consider the difference schemes applied to a derivative nonlinear system of evolution equations. For the boundary-value problem with initial conditions
 ∂u/∂t = A ∂2u/∂x2 + B ∂u/∂x + f(x,u) + g(x,u) ∂u/∂x, (t,x) ∈ (0, T] x (0, 1),u(t,0) = u(t,1) = 0, t ∈ [0, T],u(0,x) = u(0)(x), x ∈ (0, 1)
 we use the Crank-Nicolson discretization. A is complex and B – real diagonal matrixes; u, f and g are complex vector-functions. The analysis shows that proposed schemes are uniquely solvable, convergent and stable in a grid norm W22 if all (diagonal) elements in Re(A) are positive. This is true without any restriction on the ratio of time and space grid steps.

A model equation for non-resonant interacting ion acoustic plasma waves

The interaction among non-resonant ion acoustic plasma waves with different group velocities that are not close to each other is studied by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. It is shown that the nonlinear Schrödinger equation is not adequate, and instead a model system of nonlinear evolution equations is necessary to describe oscillation amplitudes of Fourier modes. This system is C-integrable, i.e. it can be linearized through an appropriate transformation of the dependent and independent variables. We demonstrate that the subclass of localized solutions gives rise to a solitonic phenomenology. These solutions propagate with the relative group velocity and maintain their shape during a collision, the only change being a phase shift. Numerical calculations confirm the validity of these predictions.

A new system of equations which predicts the evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption

The problem of the capillary-gravity waves which may arise at an interface between two stratified fluids of different densities is investigated. Particular attention is paid to the case when two different wave modes move at the same speed and to the wave train produced by the ensuing interaction. In contrast to most previous studies, the wave steepness and the wave bandwidth are not taken to be of the same order of magnitude, but the latter is of one order smaller. This leads to a system of nonlinear evolution equations which can be used to predict the subsequent progression of the wave field. These equations may be compared with the more usual nonlinear Schrödinger set which are valid under the equal bandwidth assumption and also a recently derived set which describe broader bandwidth waves. A large class of solutions to the equations is found and the corresponding wave profiles are presented.

Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations

A new algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented. The algorithm is implemented inMathematica. The programcondens.mautomatically carries out the lengthy symbolic computations for the construction of conserved densities. The code is tested on several well-known partial differential equations from soliton theory. For systems with parameters,condens.mcan be used to determine the conditions on these parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for integrability of the system.

Resonant quasiperiodic patterns in a three-dimensional lasing medium

Starting from the Maxwell-Bloch equations for a three-dimensional (3D) ring-cavity laser, we analyze stability of the nonlasing state and demonstrate that, at the instability threshold, the wave vectors of the critical perturbations belong to a paraboloid in the 3D space. Then, we derive a system of nonlinear evolution equations above the threshold. The nonlinearity in these equations is cubic. For certain sets of four spatial modes whose vectors belong to the critical paraboloid, the cubic nonlinearity gives rise to a resonant coupling between them. This is a nontrivial example of a nonlinear dissipative system in which the cubic terms are resonant. The equations for the four coupled amplitudes have two different solutions that are simultaneously stable: the single-mode one and a solution in which all the amplitudes are equal, while a certain combination of the phases is $\ensuremath{\pi}$. The latter solution gives rise to a quasiperiodic pattern in the infinite 3D cavity. We also consider effects of the boundary conditions and demonstrate that if the cavity's cross section is a trapezium it may support the quasiperiodic four-mode state rather than suppressing it. Using the Lyapunov function, we find that for the ring-laser configuration the four-mode state is metastable. However, we demonstrate that for a Fabry-P\'erot cavity, where diffusion washes out the standing-wave grating, this state is absolutely stable. We also consider a number of more complicated patterns. We demonstrate that adding a pair of resonant vectors, or any number of nonresonant ones, always produces an unstable solution. A set containing several resonant quartets without resonant coupling between them may be stable, but it is less energetically favorable than a single quartet.

Nonlinear evolution and secondary instabilities of Marangoni convection in a liquid–gas system with deformable interface

The paper presents a theory of nonlinear evolution and secondary instabilities in Marangoni (surface-tension-driven) convection in a two-layer liquid–gas system with a deformable interface, heated from below. The theory takes into account the motion and convective heat transfer both in the liquid and in the gas layers. A system of nonlinear evolution equations is derived that describes a general case of slow long-scale evolution of a short-scale hexagonal Marangoni convection pattern near the onset of convection, coupled with a long-scale deformational Marangoni instability. Two cases are considered: (i) when interfacial deformations are negligible; and (ii) when they lead to a specific secondary instability of the hexagonal convection.In case (i), the extent of the subcritical region of the hexagonal Marangoni convection, the type of the hexagonal convection cells, selection of convection patterns – hexagons, rolls and squares – and transitions between them are studied, and the effect of convection in the gas phase is also investigated. Theoretical predictions are compared with experimental observations.In case (ii), the interaction between the short-scale hexagonal convection and the long-scale deformational instability, when both modes of Marangoni convection are excited, is studied. It is shown that the short-scale convection suppresses the deformational instability. The latter can appear as a secondary long-scale instability of the short-scale hexagonal convection pattern. This secondary instability is shown to be either monotonic or oscillatory, the latter leading to the excitation of deformational waves, propagating along the short-scale hexagonal convection pattern and modulating its amplitude.

Comportamento Assintótico das Soluções de um Sistema de Equações de Evolução da Hierarquia AKNS.

Consideramos um sistema de equa çõ es de evolu çã o n ã o lineares que possui solu çõ es globais para dados iniciais pequenos. Demonstra-se que as solu çõ es s ã o assint ó ticas a solu çõ es do problema linear associado quando o tempo tende a infinito. O problema de encontrar solu çõ es n ã o lineares que sejam assint ó ticas a solu çõ es lineares dadas é investigado.