Nonlinear Dispersive Equations Research Articles

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.

On the derivation of the wave kinetic equation for NLS

AbstractA fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a largekinetic time scale$T_{\mathrm {kin}} \gg 1$and in a limiting regime where the sizeLof the domain goes to infinity and the strength$\alpha $of the nonlinearity goes to$0$(weak nonlinearity). For the cubic nonlinear Schrödinger equation,$T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$and$\alpha $is related to the conserved mass$\lambda $of the solution via$\alpha =\lambda ^2 L^{-d}$.In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particularscaling lawin which the$(\alpha , L)$limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to betwofavourable scaling laws: when$\alpha $approaches$0$like$L^{-\varepsilon +}$or like$L^{-1-\frac {\varepsilon }{2}+}$(for arbitrary small$\varepsilon $), we exhibit the wave kinetic equation up to time scales$O(T_{\mathrm {kin}}L^{-\varepsilon })$, by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales$T_*\ll T_{\mathrm {kin}}$and identify specific interactions that become very large for times beyond$T_*$. In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond$T_*$toward$T_{\mathrm {kin}}$for such scaling laws seems to require new methods and ideas.

On connection between the order of a stationary one-dimensional dispersive equation and the growth of its convective term

A boundary value problem for a stationary nonlinear dispersive equation of 2l+1 order with a convective term in the form u^ku_x, k\in N was considered on an interval (0,L). The existence, uniqueness and continuous dependence of a regular solution as well as a relation between the order l and critical values of k of the equation have been established.

Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line

In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-posedness of canonical nonlinear dispersive equations on the real line. In particular, we implement this simplified approach to an infinite iteration of normal form reductions in the context of the cubic nonlinear Schr\odinger equation (NLS) and the modified KdV equation (mKdV) on the real line and prove unconditional well-posedness in $H^s(\mathbb R)$ with (i) $s\geq \frac 16$ for the cubic NLS and (ii) $s > \frac 14$ for the mKdV. Our normal form approach also allows us to construct weak solutions to the cubic NLS in $H^s(\mathbb R)$, $0 \leq s < \frac 16$, and distributional solutions to the mKdV in $H^\frac{1}{4}(\mathbb R)$ (with some uniqueness statements).

On the work of Jean Bourgain in nonlinear dispersive equations

On the work of Jean Bourgain in nonlinear dispersive equations

A qualitative study and numerical simulations for a time-delayed dispersive equation

This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of order four. First, we prove the well-posedness of the system and give some regularity results. Then, we show that the zero solution of the system exponentially converges to zero when the time tends to infinity provided that the time-delay is small and the damping term satisfies reasonable conditions. Lastly, an intensive numerical study is put forward and numerical illustrations of the stability result are provided.

Triangulations and soliton graphs for totally positive Grassmannian

The KP equation is a nonlinear dispersive wave equation which provides an excellent model for resonant interactions of shallow-water waves. It is well known that regular soliton solutions of the KP equation may be constructed from points in the totally nonnegative GrassmannianGr(N,M)≥0. Kodama and Williams studied the asymptotic patterns (tropical limit) of KP solitons, called soliton graphs, and showed that they correspond to Postnikov's Le-diagrams. In this paper, we consider soliton graphs for the KP hierarchy, a family of commuting flows which are compatible with the KP equation. For the positive GrassmannianGr(2,M)>0, Kodama and Williams showed that soliton graphs are in bijection with triangulations of the M-gon. We extend this result to Gr(N,M)>0 when N=3 and M=6,7 and 8. In each case, we show that soliton graphs are in bijection with Postnikov's plabic graphs, which generalize Le-diagrams.

Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

A note on blow-up criteria for a class of nonlinear dispersive wave equations with dissipation

In this note, we study the Cauchy problem for a class of nonlinear dispersive wave equation with dissipative term on the real line. We establish a new local-in-space blow-up criterion. Our results improve the corresponding ones in the previous paper.

Conservative discontinuous Galerkin methods for the nonlinear Serre equations

In this paper, we develop three conservative discontinuous Galerkin (DG) schemes for the one-dimensional nonlinear dispersive Serre equations, including two conserved schemes for the equations in conservative form and a Hamiltonian conserved scheme for the equations in non-conservative form. One of the schemes owns the well-balanced property via constructing a high order approximation to the source term for the Serre equations with a non-flat bottom topography. By virtue of the Hamiltonian structure of the Serre equations, we introduce an Hamiltonian invariant and then develop a DG scheme which can preserve the discrete version of such an invariant. Numerical experiments in different cases are performed to verify the accuracy and capability of these DG schemes for solving the Serre equations.

Full Family of Flattening Solitary Waves for the Critical Generalized KdV Equation

For the critical generalized KdV equation partial _t u + partial _x (partial _x^2 u + u^5)=0 on {mathbb {R}}, we construct a full family of flattening solitary wave solutions. Let Q be the unique even positive solution of Q''+Q^5=Q. For any nu in (0,frac{1}{3}), there exist global (for tge 0) solutions of the equation with the asymptotic behavior u(t,x)=t-ν2Qt-ν(x-x(t))+w(t,x)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} u(t,x)= t^{-\\frac{\\nu }{2}} Q\\left( t^{-\\nu } (x-x(t))\\right) +w(t,x) \\end{aligned}$$\\end{document}where, for some c>0, x(t)∼ct1-2νand‖w(t)‖H1(x>12x(t))→0ast→+∞.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} x(t)\\sim c t^{1-2\\nu } \\quad \\text{ and }\\quad \\Vert w(t)\\Vert _{H^1(x>\\frac{1}{2} x(t))} \\rightarrow 0\\quad \\text{ as } t\\rightarrow +\\infty . \\end{aligned}$$\\end{document}Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.

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.

Wilton Ripples in Weakly Nonlinear Dispersive Models of Water Waves: Existence and Analyticity of Solution Branches

Traveling waves on the surface of the ocean play an important role in many oceanographic processes which necessitates a detailed quantitative understanding of their properties. The water wave equations, which govern the free-surface evolution of an ideal fluid, are the most successful model for this phenomena, but are exceedingly difficult to analyze due to their strongly nonlinear character and the fact that they are posed on a domain with moving boundary. For this reason, weakly nonlinear dispersive models are an essential tool for practitioners, and in this contribution, we study traveling wave solutions of a broad class of such models. The simplest family of traveling wave solutions are the Stokes waves which can be characterized as simple bifurcation (one-dimensional null space of the linearized operator) from the trivial (flat-water) branch of solutions. We focus our analysis on the much less studied non-simple case of Wilton ripples which have linear behavior characterized by two co-propagating harmonics (a two-dimensional null space of the linearized operator). More specifically, we show that such branches of solutions exist for a class of nonlinear dispersive model equations, and that they are analytic with respect to a natural wave height/slope parameter.

On the unique continuation of solutions to non-local non-linear dispersive equations

We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if are two suitable solutions of the equation defined in such that for some non-empty open set for then for any The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann, and some extensions obtained here.

An efficient numerical treatment for the asymptotic behaviour of the nonlinear Airy-type problems

This study focuses on symplectic integrators for numerical evaluation of the asymptotic solutions of the nonlinear Airy-type equations obtained by reducing the nonlinear dispersive equations. Since the nature of Airy-type equations has both highly oscillatory slow decay and exponential fast decay, most of classical integrators are not able to correctly exhibit challenging physical behaviour. We use specially designed symplectic integrators combining splitting methods with Magnus integrators to catch asymptotic behaviour of nonlinear Airy-type equations efficiently, even for large step sizes. Efficiency of the proposed methods for given problems is discussed. Moreover, numerical results obtained by the proposed methods are compared with the existing results in the literature.

Global Existence and Singularity of the N-Body Problem with Strong Force

We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schrodinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. We are able to give a conditional dichotomy of the global existence and singularity below the excited energy in Theorem 4, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function $$K_\omega $$. The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For $$N\ge 3$$, we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of $$K_\omega $$.

Construction of lump-type and interaction solutions to an extended (3+1)-dimensional Jimbo–Miwa-like equation

We present lump-type solutions and interaction solutions to an extended (3[Formula: see text]+[Formula: see text]1)-dimensional Jimbo–Miwa-like equation. Three classes of lump-type solutions are obtained by the Hirota bilinear method. Interaction solutions are among lump-type solutions, two kink waves and periodic waves, and between two kink waves and a periodic wave are computed. Dynamical characters of the obtained solutions are graphically exhibited. These wave solutions enrich the dynamical theory of higher-dimensional nonlinear dispersive wave equations.

The solitary wave solutions of the nonlinear perturbed shallow water wave model

In this paper, we study the nonlinear perturbed shallow water wave model, which satisfies the asymptotic integrability condition with this family: the KdV equation, Camassa–Holm equation and Degasperis–Procesi equation. We establish the existence of solitary wave solutions for the perturbed nonlinear dispersive equations by means of the geometric singular perturbation and invariant manifold theory.

Nonlinear sub-diffusion and nonlinear sub-diffusion dispersion equations and their proposed solutions

Abstract Many investigations related to the analytical solutions of the nonlinear sub-diffusion equation exist. In this paper, we investigate the conditions under which the analytical and the approximate solutions of the nonlinear sub-diffusion equation and the nonlinear sub-advection dispersion equation exist. In other words, the problems of existence and uniqueness of the solutions the fractional diffusion equations have been addressed. We use the Banach fixed Theorem. After proving the existence and uniqueness, we propose the analytical and the approximate solutions of the nonlinear sub-diffusion, and the nonlinear sub-advection dispersion equations. We analyze the impact of the sub-diffusion coefficient, the advection coefficient and the dispersion coefficient in the diffusion processes. The homotopy perturbation Laplace transform method has been used in this paper. Some numerical examples are provided to illustrate the main results of the article.

Successive Approximation Method for Solving Wu-Zhang Systems of (1 1) Dimensional

In this section the Successive approximate method (S.A.M) introduced for solving the Wu-Zhang systems, a (1 + 1)-dimensional nonlinear dispersive wave equation, this method shows us that the technique provided without disorder, in this model of convergence power series with a simple calculated ingredients and gives effective results.