We consider periodic solutions to equations of Korteweg-Devries type. While the stability theory for periodic waves has received much some attention the theory is much less developed than the analogous theory for solitary wave stability, and appears to be mathematically richer. We prove an index theorem giving an exact count of the number of unstable eigenvalues of the linearized operator in terms of the number of zeros of the derivative of the traveling wave profile together with geometric information about a certain map between the constants of integration of the ordinary differential equation and the conserved quantities of the partial differential equation.
This index can be regarded as a generalization of both the Sturm oscillation theorem and the classical stability theory for solitary wave solutions for equations of Korteweg-de Vries type. In the case of a polynomial nonlinearity this index, together with a related one introduced earlier by Bronski and Johnson, can be expressed in terms of derivatives of period integrals on a Riemann surface. Since these period integrals satisfy a Picard-Fuchs equation these derivatives can be expressed in terms of the integrals themselves, leading to an expression in terms of various moments of the solution. We conclude with some illustrative examples.
We prove that if a solution of an equation of KdV type is bounded above by a traveling wave with an amplitude that decays faster than a given linear exponential then it must be zero. We assume no restrictions neither on the size nor in the direction of the speed of the traveling wave.
We prove that the Novikov-Veselov equation (an analog of KdV in dimension 2 + 1) at zero energy does not have sufficiently localized soliton solutions of conductivity type.
We demonstrate that the Novikov.Veselov equation (a (2+1)-dimensional analog of KdV) at zero energy does not possess solitons with the space localization stronger than O(|x|^{-4}).
In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation $$u_t+uu_x+u_{xxx}-γ_1 u_{xxxxx}=0,$$ and the modified Kawahara-KdV equation $$u_t+3u^2u_x+u_{xxx}-γ_2 u_{xxxxx}=0,$$ where $γ_i\in\mathbb{R}$ is a positive number when $i=1,2$. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by Albert
We analyze the stationary and traveling wave solutions to a family of degenerate dispersive equations of KdV and NLS-type. In stark contrast to the standard soliton solutions for non-degenerate KdV and NLS equations, the degeneracy of the elliptic operators studied here allows for compactly supported steady or traveling states. As we work in $1$ dimension, ODE methods apply, however the models considered have formally conserved Hamiltonian, Mass and Momentum functionals, which allow for variational analysis as well.
For $N\leq34,$ we construct traveling waves with small speed for the Gross-Pitaevskii equation, by gluing $N(N+1)/2$ pairs of degree $\pm1$ vortices of the Ginzburg-Landau equation. The location of these vortices is symmetric in the plane and determined by the Adler-Moser polynomials, which has its origin in the study of Calogero-Moser system and rational solutions of the KdV equation. The construction still works for $N>34$, under the additional assumption that the corresponding Adler-Moser polynomial has no repeated root. It is expected that this assumption holds for any $N\in\mathbb{N}$.
We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data.
We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.
We prove special decay properties of solutions to the initial value problem associated to the $k$-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As application of our method we also obtain results concerning the decay behavior of perturbations of the traveling wave solutions as well as results for solutions corresponding to special data.
The Hamiltonian-Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem $ J L u=λu$, where $J$ is skew-symmetric and $L$ is self-adjoint. If $J$ has a bounded inverse the index is well-established, and it is given by the number of negative eigenvalues of the operator $L$ constrained to act on some finite-codimensional subspace. There is an important class of problems - namely, those of KdV-type - for which $J$ does not have a bounded inverse. In this paper we overcome this difficulty and derive the index f…
The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.
We study the existence of solitary waves in a diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. For monatomic FPUT the traveling wave equations are a regular perturbation of the Korteweg-de Vries (KdV) equation's but, surprisingly, we find that for the diatomic lattice the traveling wave equations are a singular perturbation of KdV's. Using a method first developed by Beale to study traveling solutions for capillary-gravity waves we demonstrate that for wave speeds in slight excess of the lattice's speed of sound there exists nontrivial traveling wave solutions which are the superposition an e…
We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDE of KdV-type, including generalized KdV and Benjamin-Ono equations. In this investigation, we consider the spectral stability of such solutions that arise as small perturbations of an equilibrium state. A key feature of our analysis is the development of a nonlocal Floquet-like theory that is suitable to analyze the $L^2(\RM)$ spectrum of the associated linearized operators. Using spectral perturbation theory then, we derive a relationship between the power of th…
We prove the existence of a global bifurcation branch of $2π$-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class $C^α$, $α< \frac{1}{2}$. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning …
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work \cite{carles}, in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in \cite{natali1} to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in \cite{grillakis1} and \cite{weinstein1} to deduce the orbital stability of the periodic t…
This paper sheds new light on the stability properties of solitary wave solutions associated with models of Korteweg-de Vries and Benjamin\&Bona\&Mahoney type, when the dispersion is very lower. Via an approach of compactness, analyticity and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of linear instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and linear instability of the ground states solutions for both models is obtained for som…
We prove that the KdV-Burgers is globally well-posed in $ H^{-1}(\T) $ with a solution-map that is analytic from $H^{-1}(\T) $ to $C([0,T];H^{-1}(\T))$ whereas it is ill-posed in $ H^s(\T) $, as soon as $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\T) $ to even ${\cal D}'(\T) $ at any fixed $ t>0 $ small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dissipation part of the KdV-Burgers equation allows to lower the $ C^\infty $ critical index with respect to the KdV equation, it does not permit to imp…
In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDE's and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of $p$. Namely, we show that all waves are spectrally stable for $2<p\leq 8$, while a single mode in…
Given a suitable solution $V(t,x)$ to the Korteweg--de Vries equation on the real line, we prove global well-posedness for initial data $u(0,x) \in V(0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles $V(0,x) \in H^5(\mathbb{R}/\mathbb{Z})$ satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In our companion paper we show that smooth step-like initial data also satisfy our hypotheses. …
Smooth periodic travelling waves in the Camassa--Holm (CH) equation are revisited. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common to the Korteweg--de Vries (KdV) equation, has several disadvantages, e.g., the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We explore the nonstandard formulation and prove that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a…
We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evan's function) in a neighborhood of the origin in the spectral plane leads …
Scalar conservation laws with non-convex fluxes have shock wave solutions that violate the Lax entropy condition. In this paper, such solutions are selected by showing that some of them have corresponding traveling waves for the equation supplemented with dissipative and dispersive higher-order terms. For a cubic flux, traveling waves can be calculated explicitly for linear dissipative and dispersive terms. Information about their existence can be used to solve the Riemann problem, in which we find solutions for some data that are different from the classical Lax-Oleinik construction. We consi…
A unified geometric approach for the stability analysis of traveling pulse solutions for reaction-diffusion equations with skew-gradient structure has been established in a previous paper [9], but essentially no results have been found in the case of traveling front solutions. In this work, we will bridge this gap. For such cases, a Maslov index of the traveling wave is well-defined, and we will show how it can be used to provide the spectral information of the waves. As an application, we use the same index providing the exact number of unstable eigenvalues of the traveling front solutions of…
Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \quad v(x,0)=v_0(x)\] posed on the torus $\mathbb{T}$ in the spaces \[ {\cal H}^s_1:=H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_2:=H^s_0 (\mathbb{T})\times H^s(\mathbb{T}), \quad {\cal H}^s_3:=H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_4:=H^s (\mathbb{T})\times H^s (\mathbb{T}).\] For $k=1,2,3,4$, it is…
We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal Teichmueller space, is endowed with a right-invariant Kaehler metric. Using results from the theory of quasiconformal mappings we construct an embedding of T into the infinite dimensional Segal-Wilson Grassmannian. The latter turns out to be a very natural ambient space for T. This allows us to prove that T's sectional curvature is negative in the holomorphic d…
We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schrödinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear…
We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass $\wp$ functions of both third and fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV--BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstra…
This paper concerns spectral stability of nonlinear waves in KdV-type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and an unbounded non-invertible symplectic operator $\partial_x$. The instability index theorem is proven under a generic assumption on the self-adjoint operator both in the case of solitary waves and periodic waves. This result is reviewed in the context of other recent results on spectral stability of nonlinear waves in KdV-type evolution equations.
We study the existence of traveling waves in a spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. This is a one-dimensional lattice of identical particles connected by alternating nonlinear springs. Following the work of Faver and Wright on the mass dimer, or diatomic, lattice, we find that the lattice equations in the long wave regime are singularly perturbed and apply a method of Beale to produce nanopteron traveling waves with wave speed slightly greater than the lattice's speed of sound. The nanopteron wave profiles are the superposition of an exponentially decaying term (which itself i…
In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable struc…
Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full governing equations by taking into account a prescribed but arbitrary depth-dependent density distribution. In contrast to the case of constant density, we obtain for each fixed water depth a different model equation for the horizontal component of the velocity field. We derive explicit formulas for traveling wave solutions of these model equations and perform a de…
We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the KdV line solitary wave or the Zaitsev solitary waves which are localized in $x$ and $y$ periodic or conversely).
We prove real analyticity of all the streamlines, including the free surface, of a steady stratified flow of water over a flat bed in the absence of stagnation points, with a Hölder continuous Bernoulli function and a Hölder continuously differentiable density function. Furthermore, we show that if the Bernoulli function and the density function possess some Gevrey regularity of index s, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals to 1, then we obtain analyticity of the stream function. The regularity result…
We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index $s$, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index $s$ equals to 1, then we obtain analyticity of the stream function. The…
Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman-Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for anal…
In this paper, we investigate the instability of one-dimensionally stable periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Zakharov-Kuznetsov equation in two space dimensions. By deriving appropriate asymptotic expansions of the periodic Evans function, we derive an index which yields sufficient conditions for transverse instabilities to occur. This index is geometric in nature, and applies to any periodic traveling wave profile under some minor smoothness assumptions on the nonlinearity. We also desc…
We study traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattices in the long wave limit. Such lattices are known to possess nanopteron traveling waves in relative displacement coordinates. These nanopteron profiles consist of the superposition of an exponentially localized "core," which is close to a KdV solitary wave, and a periodic "ripple," whose amplitude is small beyond all algebraic orders of the long wave parameter, although a zero amplitude is not precluded. Here we deploy techniques of spatial dynamics, inspired by results of Iooss and Kirchgässner, Iooss and…
Planar Gross--Pitaevskii traveling waves at every subsonic speed
University of Macau（澳门大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；The Chinese University of Hong Kong（香港中文大学）
We establish instability of periodic traveling waves arising in conservation laws featuring phase transition. The analysis uses the Evans function framework introduced by R.A. Gardner in the periodic case. The main new tool is a periodic generalization of the stability index introduced by Gardner and Zumbrun in the traveling front or pulse case.
