Nonlinear Dispersive Partial Differential Equations Research Articles

Local existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the p-generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions:c×⋯×c︸p(total distance):=∑♣1,⋯,♣p∈Zν♣1+⋯+♣p=total distance∏j=1pc(♣j). In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for p=2[11], and the combinatorics become harder for larger values of p. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, p=3. Next, we prove existence and uniqueness in the general case p≥2, which then covers the remaining cases p≥4. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general p.

Testing by wave packets and modified scattering in nonlinear dispersive pde’s

Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely. An interesting example is that of problems with cubic nonlinearities in one space dimension. The method of testing by wave packets was introduced by the authors as a tool to efficiently capture the asymptotic equations associated to such flows, and thus establish the modified scattering mechanism in a simpler, more efficient fashion, and at lower regularity. In these expository notes we describe how this method can be applied to problems with general dispersion relations.

A Brief Historical Overview and Mathematical Equations of Soliton

bstract: A soliton is a strongly stable, nonlinear, self-reinforcing, localised wave packet that maintains its shape even after colliding with other similar localised wave packets and propagates freely at a constant speed. The medium's balanced cancellation of nonlinear and dispersive effects is responsible for its extraordinary stability. Dispersive effects are a characteristics of several systems where a wave frequency determines how fast it travels. Subsequently, it was shown that soliton solutions offer stable solutions for a broad range of weakly nonlinear dispersive partial differential equations that characterise physical systems. Solitons have a self-restoring quality that makes them desirable for long-distance, high-speed transmissions. The strain on networks is quickly reaching a critical threshold as transmission speeds over longer distances are now approaching 40 Gbit/sec. In the future, solitons might offer intriguing alternatives.

Generalized compacton equation, conservation laws and exact solutions

Compactons were introduced as solutions of nonlinear dispersive partial differential equations. There exists a variety of third-order equations that support compacton solutions. We consider a generalized third-order nonlinear K(fm,gn) equation that generalizes Rosenau–Hyman K(m,n) equation, extended Rosenau–Pikovsky K(cosmu,cosnu) equation, logarithmic KdV equation, generalized Gardner, and several other third-order nonlinear dispersive equations. In this paper, we obtain conservation laws of the K(fm,gn) equation in the most generalized form of f(u) and g(u). Several exact solutions of various equations of the K(fm,gn) form are derived from these generalized conservation laws by using the double reduction theory.

A fifth-order dispersive partial differential equation for curve flow on the sphere

The initial value problem for a fifth-order nonlinear dispersive partial differential equation describing the curve flow on the sphere is considered. A typical example of the equation arises in a hierarchy of completely integrable systems containing one-dimensional classical Heisenberg ferromagnetic spin model. This paper establishes the local existence and uniqueness of a solution to the initial value problem under the periodic boundary condition. The proof is based on the energy method combined with a kind of gauge transformation to overcome the difficulty of a loss of derivative.

Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE

In this paper, we consider the periodic Cauchy problem for a stochastically perturbed nonlinear dispersive partial differential equation with cubic nonlinearity, which involves the integrable Novikov equation arising from the shallow water wave theory as a special case. We first establish the existence and uniqueness of local pathwise solutions in Sobolev spaces Hs(T)(s>32) with nonlinear multiplicative noise, where the key ingredients are the stochastic compactness method, the Skorokhod representation theorem and the Gyöngy–Krylov characterization of convergence in probability. In the case of linear multiplicative noise, we investigate the conditions which lead to the blow-up phenomena and global existence of pathwise solution. Finally, we show that the linear multiplicative noise has a dissipative effect on the periodic peakon solutions to the associated deterministic Novikov equation.

Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application

A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.

Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative.

In this paper, the time-fractional nonlinear dispersive (TFND) partial differential equations (PDEs) in the sense of conformable fractional derivative (CFD) are proposed and analyzed. Three types of TFND partial differential equations are considered in the sense of CFD, which are the TFND Boussinesq, TFND Klein-Gordon, and TFND B(2, 1, 1) PDEs. Solitary pattern solutions for this class of TFND partial differential equations based on the residual fractional power series method is constructed and discussed. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions. The results suggest that the algorithm presented here offers solutions to problems in a rapidly convergent series leading to ideal solutions. Furthermore, the results obtained are like those in previous studies that used other types of fractional derivatives. In addition, the calculations used were much easier and shorter compared with other types of fractional derivatives.

A new non-polynomial spline method for solution of linear and non-linear third order dispersive equations

In this paper, a new three-level implicit method is developed to solve linear and non-linear third order dispersive partial differential equations. The presented method is obtained by using exponential quartic spline to approximate the spatial derivative of third order and finite difference discretization to approximate the first order spatial and temporal derivative. The developed method is tested on four examples and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, the truncation error and stability analysis of the presented method are investigated, and graphical comparison between analytical and approximate solution is also shown for each example.

On solutions of the fifth-order dispersive equations with porous medium type non-linearity

In this work, we focus on obtaining the exact solutions of the fifth-order semi-linear and non-linear dispersive partial differential equations, which have the second-order diffusion-like (porous-type) non-linearity. The proposed equations were not studied in the literature in the sense of the exact solutions. We reveal solutions of the proposed equations using the classical Riccati equations method. The obtained exact solutions, which can play a key role to simulate non-linear waves in the medium with dispersion and diffusion, are illustrated and discussed in details.

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

The Kolmogorov–Zakharov Model for Optical Fiber Communication

A mathematical framework is presented to study the evolution of multi-point cumulants in nonlinear dispersive partial differential equations with random input data, based on the theory of weak wave turbulence (WWT). This framework is used to explain how energy is distributed among Fourier modes in the nonlinear Schr\"odinger equation. This is achieved by considering interactions among four Fourier modes and studying the role of the resonant, non-resonant, and trivial quartets in the dynamics. As an application, a power spectral density is suggested for calculating the interference power in dense wavelength-division multiplexed optical systems, based on the kinetic equation of the WWT. This power spectrum, termed the Kolmogorov-Zakharov (KZ) model, results in a better estimate of the signal spectrum in optical fiber, compared with the so-called Gaussian noise (GN) model. The KZ model is generalized to non-stationary inputs and multi-span optical systems.

Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation.

The Hasimoto transformation between the classical LIA (local induction approximation, a model approximating the motion of a thin vortex filament) and the nonlinear Schrödinger equation (NLS) has proven very useful in the past, since it allows one to construct new solutions to the LIA once a solution to the NLS is known. In the present paper, the quantum form of the LIA (which includes mutual friction effects) is put into correspondence with a type of complex nonlinear dispersive partial differential equation (PDE) with cubic nonlinearity (similar in form to a Ginsburg-Landau equation, with additional nonlinear terms). Transforming the quantum LIA in such a way enables one to obtain quantum vortex filament solutions once solutions to this dispersive PDE are known. From our quantum Hasimoto transformation, we determine the form and behavior of Stokes waves, a standing one-soliton, traveling waves, and similarity solutions under normal and binormal friction effects. The quantum Hasimoto transformation is useful when normal fluid velocity is relatively weak, so for the case where the normal fluid velocity is dominant we resort to other approaches. We exhibit a number of solutions that exist only in the presence of the normal fluid velocity and mutual friction terms (which would therefore not exist in the limit taken to obtain the classical LIA, decaying into line filaments under such a limit), examples of which include normal fluid driven helices, stationary and propagating topological solitons, and a vortex ring whose radius varies inversely with the normal fluid magnitude. We show that, while chaos may not be impossible under the quantum LIA, it should not be expected to arise from traveling waves along quantum vortex filaments under the quantum LIA formulation.

Fractal solutions of linear and nonlinear dispersive partial differential equations

In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schr\"odinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schr\"odinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac32$ for which the evolution of the curvature is given by a periodic nonlinear Schr\"odinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang {\em et al.} \cite{csu} and Nahmod {\em et al.} \cite{nsvz}.

Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations

Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.

General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation

In his study of superfluid turbulence in the low-temperature limit, Svistunov [“Superfluid turbulence in the low-temperature limit,” Phys. Rev. B 52, 3647 (1995)] derived a Hamiltonian equation for the self-induced motion of a vortex filament. Under the local induction approximation (LIA), the Svistunov formulation is equivalent to a nonlinear dispersive partial differential equation. In this paper, we consider a family of rotating vortex filament solutions for the LIA reduction of the Svistunov formulation, which we refer to as the 2D LIA (since it permits a potential formulation in terms of two of the three Cartesian coordinates). This class of solutions holds the well-known Hasimoto-type planar vortex filament [H. Hasimoto, “Motion of a vortex filament and its relation to elastica,” J. Phys. Soc. Jpn. 31, 293 (1971)] as one reduction and helical solutions as another. More generally, we obtain solutions which are periodic in the space variable. A systematic analytical study of the behavior of such solutions is carried out. In the case where vortex filaments have small deviations from the axis of rotation, closed analytical forms of the filament solutions are given. A variety of numerical simulations are provided to demonstrate the wide range of rotating filament behaviors possible. Doing so, we are able to determine a number of vortex filament structures not previously studied. We find that the solution structure progresses from planar to helical, and then to more intricate and complex filament structures, possibly indicating the onset of superfluid turbulence.

Numerical study of blow-up in the Davey-Stewartson system

Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow up. We numerically study the long time behavior and potential blow-up of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa solutions. It is shown in this way that both are unstable to blow-up and dispersion, and that blow-up in the Ozawa solution is generic.

On the Cauchy problem for the modified Novikov equation with peakon solutions

A new nonlinear dispersive partial differential equation with cubic nonlinearity, which includes the famous Novikov equation as special case, is investigated. We first establish the local well-posedness in a range of the Besov spaces Bp,rs, p,r∈[1,∞], s>max{32,1+1p} but s≠2+1p (which generalize the Sobolev spaces Hs), well-posedness in Hs with s>32, is also established by applying Katoʼs semigroup theory. Then we give the precise blow-up scenario. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that peakon solutions to the equation are global weak solutions.

A model containing both the Camassa–Holm and Degasperis–Procesi equations

A nonlinear dispersive partial differential equation, which includes the famous Camassa–Holm and Degasperis–Procesi equations as special cases, is investigated. Although the H 1 -norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space H s with 1 < s ⩽ 3 2 is established under the assumptions u 0 ∈ H s and ‖ u 0 x ‖ L ∞ < ∞ . The local well-posedness of solutions for the equation in the Sobolev space H s ( R ) with s > 3 2 is also developed.

Gevrey regularity for a class of water-wave models

Local well posedness for a class of higher-order nonlinear dispersive partial differential equations is obtained in spaces of functions analytic on a strip around the real axis. The proof relies on estimates in space–time norms adapted to the linear part of the equation. The class of equations in view contains a number of equations arising in the modeling of waves in fluids and also in other applications.