Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms

We study the (n + 1)-dimensional generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation, a universal equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in n + 1 dimensions, and arising in several physical contexts, such as acoustics, plasma physics and hydrodynamics. For n = 2, this equation is integrable, and has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. We construct an exact solution of the (n + 1)-dimensional model containing an arbitrary function of one variable, corresponding to its parabolic invariance, describing waves, constant on their paraboloidal wave front, breaking simultaneously in all points of it. Then, we use such a solution to build a uniform approximation of the solution of the Cauchy problem, for small and localized initial data, showing that such a small and localized initial data evolving according to the (n + 1)-dimensional dKP equation break, in the long time regime, if and only if 1 ⩽ n ⩽ 3, i.e., in physical space. Such a wave breaking takes place, generically, in a point of the paraboloidal wave front, and the analytic aspects of it are given explicitly in terms of the small initial data.

We study the generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation in dimensions and with nonlinearity of degree , a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if . Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [], generalize those obtained in [] for the dKP equation in dimensions with quadratic nonlinearity, and are obtained following the same strategy.

Author(s): Harrop-Griffiths, Benjamin; Ifrim, Mihaela; Tataru, Daniel | Abstract: We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets.

Generic wave train solutions to the complex Ablowitz‐Ladik equations are developed using methods of algebraic geometry. The inverse spectral transform is used to realize these solutions as potentials in a spatially discrete linear operator. The manifold of wave trains is infinite‐dimensional, but is stratified by finite‐dimensional submanifolds indexed by nonnegative integers g. Each of these strata is a foliation whose leaves are parametrized by the moduli space of (possibly singular) hyperelliptic Riemann surfaces of genus g. The generic leaf is a g‐dimensional complex torus. Thus, each wave train is constructed from a finite number of complex numbers comprising a set of spectral data, indicating that the wave train has a finite number of degrees of freedom. Our construction uses a new Lax pair differing from that originally given by Ablowitz and Ladik. This new Lax pair allows a simplified construction that avoids some of the degeneracies encountered in previous analyses that make use of the original discretized AKNS Lax pair. Generic wave trains are built from Baker‐Akhiezer functions on nonsingular Riemann surfaces having distinct branch points, and the construction is extended to handle singular Riemann surfaces that are pinched off at a coinciding pair of branch points. The corresponding solutions in the pinched case may also be derived from wave trains belonging to nonsingular surfaces using Bäcklund transformations. The problem of reducing the complex Ablowitz‐Ladik equations to the focusing and defocusing versions of the discrete nonlinear Schrödinger equation is solved by specifying which spectral data correspond to focusing or defocusing potentials. Within the class of finite genus complex potentials, spatially periodic potentials are isolated, resulting in a formula for the solution to the spatially periodic initial‐value problem. Formal modulation equations governing slow evolution of (g + 1)‐phase wave trains are developed, and a gauge invariance is used to simplify the equations in the focusing and defocusing cases. In both of these cases, the modulation equations can be either hyperbolic (suggesting modulational stability) or elliptic (suggesting modulational instability), depending upon the local initial data. As has been shown to be the case with modulation equations for other integrable systems, hyperbolic data will remain hyperbolic under the evolution, at least until infinite derivatives develop.

We consider the vorticity formulation of the two-dimensional viscous Camassa–Holm equations in the whole space. We establish global existence for solutions corresponding to initial data in $L^1$ and describe the large time behavior of solutions with sufficiently small and localized initial data. We calculate the rate at which such solutions approach an “unfiltered” Oseen vortex by computing the rate at which the solution of a scaled vorticity problem approaches the solution to a corresponding linearized equation.

<p style='text-indent:20px;'>We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.

The first target of this article is the local well-posedness question for 1D quasilinear Schrödinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a sharp local well-posedness result. The second goal of this article is to consider the long-time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: “Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions”; the conjecture was initially proved for a class of semilinear Schrödinger type models. Our work here establishes the above conjecture for 1D quasilinear Schrödinger flows. Precisely, we show that if the problem has phase rotation symmetry and is conservative and defocusing, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows where no localization condition is imposed on the data. Furthermore, we prove the global well-posedness at the minimal Sobolev regularity as in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that small, $\epsilon $ ϵ size data yields long-time solutions on the $\epsilon ^{-8}$ ϵ − 8 time-scale. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schrödinger flows, also at minimal regularity.

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

Global solutions of quasi-linear Hamiltonian mKdV equation

We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e.~nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture the difference in velocities. In this paper we present an alternative method, which is better applicable to multiple components. Our method involves a nonlinear iteration scheme that employs $L^1$-$L^p$ estimates in Fourier space and exploits oscillations in time and frequency, which arise due to differences in transport. Under the assumption that each component exhibits different velocities, we establish global existence and decay for small, algebraically localized initial data in multi-component reaction-diffusion-advection systems allowing for cubic mixed-terms and nonlinear terms of Burgers' type.

Under the internal dissipative condition, the Cauchy problem for inhomogeneous quasilinear hyperbolic systems with small initial data admits a unique global C1 solution, which exponentially decays to zero as t → +∞, while if the coefficient matrix Θ of boundary conditions satisfies the boundary dissipative condition, the mixed initial-boundary value problem with small initial data for quasilinear hyperbolic systems with nonlinear terms of at least second order admits a unique global C1 solution, which also exponentially decays to zero as t → +∞. In this paper, under more general conditions, the authors investigate the combined effect of the internal dissipative condition and the boundary dissipative condition, and prove the global existence and exponential decay of the C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with small initial data. This stability result is applied to a kind of models, and an example is given to show the possible exponential instability if the corresponding conditions are not satisfied.

Global solution for the 3D gravity water waves system above a flat bottom

Global existence and decay in nonlinearly coupled reaction-diffusion-advection equations with different velocities