In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation \[ i \sum_{j=0}^{3} \ga^j \pd_j ψ- mψ+ F(\barψψ)ψ=0 \] where $ m >0$ and $ F(s) = |s|^θ$ for $ 1\leq θ< 2.$ More precisely we show that there exists $\e_0 > 0$ such that for $ω\in(m - \e_0, m), $ there exists a solution $ ψ(t,x) = e^{-iωt}ϕ_ω(x), x_0 = t, x = (x_1, x_2, x_3),$ and the mapping from $ ω$ to $ ϕ_ω $ is continuous. We prove this result by relating the stationary solutions to the ground states of nonlinear Schrödinger equations.
This paper is concerned with the nonlinear Schrodinger lattice with nonlinear hopping. Via variation approach and the Nehari manifold argument, we obtain two types of solution: periodic ground state and localized ground state. Moreover, we consider the convergence of periodic solutions to the solitary wave.
We prove the existence of a $L^2$-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.
We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction (the Soler model) and the Dirac--Klein--Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing. We show the relation of $\pm 2ω\mathrm{i}$ eigenvalues of the linearization at a solitary wave, Bogoliubov $\mathbf{SU}(1,1)$ symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary …
The global attraction is established for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Klein-Gordon equation in one dimension coupled to a finite number of nonlinear oscillators: We prove that {\it each finite energy solution} converges as $t\to\pm\infty$ to the set of all ``nonlinear eigenfunctions'' of the form $ϕ(x)e\sp{-iωt}$ if all oscillators are strictly nonlinear, and the distances between all neighboring oscillators are sufficiently small. The {\it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum…
The long-time asymptotics is analyzed for all finite energy solutions to a model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a point. Our main result is that each finite energy solution converges as $t\to\pm\infty$ to the set of ``nonlinear eigenfunctions'' $ψ(x)e\sp{-iωt}$.
We use the perturbation theory to build solitary wave solutions $ϕ_ω(x)e^{-iωt}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\barψψ)βψ$, with $f(τ)=|τ|^k+o(|τ|^k)$, $k>0$, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit $ω\lesssim m$; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schrödinger charge-critical, one has $Q'(ω)<0$ for $ω\lesssim m$, i…
We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system proving the existence of solutions in which the Dirac wave function is of the form $ϕ(x,ω)e^{-iωt}$, $ω\in(-m,ω_*)$, with some $ω_*>-m$, such that $ϕ_ω\in H^1(\mathbb{R}^3,\mathbb{C}^4)$, $\Vertϕ_ω\Vert^2_{L^2}=O(m-|ω|)$, and $\Vertϕ_ω\Vert_{L^\infty}=O(m-|ω|)$. The method of proof is an implicit function theorem argument based on an identification of the nonrelativistic limit as the ground state of the Choquard equation.
We study the point spectrum of the linearization at a solitary wave solution $ϕ_ω(x)e^{-\mathrm{i}ωt}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(ψ^*βψ)βψ$ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with nonzero real part, in the non-relativistic limit $ω\lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(τ)=|τ|^k+O(|τ|^K)$ for $τ\to 0$, with $0<k<K$. For $n\ge 1$, we prove the spectral stability of small amplitude solitary waves ($ω\lesssim m$) for the charge-subcr…
We study the orbital instability of solitary waves for a generalized derivative nonlinear Schrödinger equation. We give sufficient conditions for instability of a two-parameter family of solitary waves in a degenerate case.
In this paper, we study a compound Korteweg-de Vries-Burgers equation with a higher-order nonlinearity. A class of solitary wave solutions is obtained by means of a series expansion.
We prove the existence of a new type of solutions to a nonlinear Schrödinger system. These solutions, which we call "multi-speeds solitary waves", are behaving at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backwards in time and using energy methods to obtain uniform estimates.
We study the nonrelativistic limit of solitary waves from Nonlinear Maxwell-Klein-Gordon equations (NMKG) to Nonlinear Schrodinger-Poisson equations (NSP). It is known that the existence or multiplicity of positive solutions depends on the choices of parameters the equations contain. In this paper, we prove that for a given positive solitary wave of NSP, which is found in Ruiz's work \cite{R}, there corresponds a family of positive solitary waves of NMKG under the nonrelativistic limit. Notably, our results contain a new result of existence of positive solutions to (NMKG) with lower order nonl…
We consider a system of coupled nonlinear Schr{ö}dinger equations in one space dimension. First, we prove the existence of multi-speed solitary waves, i.e solutions to the system with each component behaving at large times as a solitary wave. Then, we investigate numerically the interaction of two solitary waves supported each on one component. Among the possible outcomes, we find elastic and inelastic interactions, collision with mass extraction and reflexion.
We numerically study solitary waves in the coupled nonlinear Schrödinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change i…
We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.
We consider the nonlinear Dirac equation, also known as the Soler model: $i\p\sb tψ=-iα\cdot \nabla ψ+m βψ-f(ψ\sp\ast βψ) βψ$, $ψ(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, $f\in C\sp 2(\R)$, where $α_j$, $j = 1,...,n$, and $β$ are $N \times N$ Hermitian matrices which satisfy $α_j^2=β^2=I_N$, $α_j β+βα_j=0$, $α_j α_k + α_k α_j =2 δ_{jk} I_N$. We study the spectral stability of solitary wave solutions $ϕ(x)e^{-iωt}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $ω\to m$, proving that if $k>2/n$, then one positive…
The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group. For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a $\textit{fundamental solitary wave}$. By using a result of one of the authors and his collaborator, the bifurcations of the fundamental solitary wave are detected. We utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral or orbital stability of the bifurcated solitary waves as well as as that of the fundamental one…
This paper proves existence and stability results of solitary-wave solutions to coupled nonlinear Schrödinger equations with power-type nonlinearities arising in several models of modern physics. The existence of solitary waves is obtained by solving a variational problem subject to two independent constraints and using the concentration-compactness method. The set of minimizers is shown to be stable and further information about the structures of this set are given. The paper extends the results previously obtained by Cipolatti and Zumpichiatti, Nguyen and Wang, and Ohta.
We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1d model and the transverse perturbation "have the same sign". Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov-Kuznetsov equation and we hope that it may be useful in other contexts.
We prove the existence of travelling-wave solutions for a system of coupled nonlinear Schrödinger equations arising in nonlinear optics. Such a system describes second-harmonic generation in optical materials with $χ^{(2)}$ nonlinearity. To prove the existence of travelling waves, we employ the method of concentration compactness to prove the relative compactness of minimizing sequences of the associated variational problem.
We show existence of solitary-wave solutions to the equation \begin{equation*} u_t+ (Lu - n(u))_x = 0\,, \end{equation*} for weak assumptions on the dispersion $L$ and the nonlinearity $n$. The symbol $m$ of the Fourier multiplier $L$ is allowed to be of low positive order ($s > 0$), while $n$ need only be locally Lipschitz and asymptotically homogeneous at zero. We shall discover such solutions in Sobolev spaces contained in $H^{1+s}$.
We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, their velocity. We illustrate these results with numerical simulations.
We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2α\partial_{t}u-Δu+u-|u|^{p-1}u=0 \end{equation*} on $\bf R^N$, for $1\leq N\leq 5$ and energy subcritical exponents $p>2$. The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solitary waves in the case of opposite signs. Close to the sum of two remote solitary waves, it turns out that only the components of the initial data in the unstable direction of…
In this paper, we study the existence of positive solutions to the nonlinear elliptic system, which is derived from taking the nonrelativistic limit of the nonlinear Maxwell-Klein-Gordon equations under the decomposition of waves functions into positron and electron parts. We characterize the existence and nonexistence of positive vector solutions, depending on parameters $p$ and $μ_{ij}$.
We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2σ}\partial_x u=0, $$ where $1<σ<2$. The equation has a two-parameter family of solitary wave solutions of the form $$ u_{ω,c}(t,x)=e^{iωt+i\frac c2(x-ct)-\frac{i}{2σ+2}\int_{-\infty}^{x-ct}φ^{2σ}_{ω,c}(y)dy}φ_{ω,c}(x-ct). $$ The stability theory in the frequency region of $|c|<2\sqrtω$ was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case $c=2\sqrtω$.
Nonlinear Schrödinger (NLS) equations with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves, and to the long-time dynamics of solutions of (NLS). We study these spectra in detail, both analytically and numerically.
In this paper we study a class of nonlinear Schrödinger equations which admit families of small solitary wave solutions. We consider solutions which are small in the energy space $H^1$, and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we show that as $t \to \infty$, the solitary wave component converges to a fixed solitary wave, and the dispersive component converges to a solution of the free Schrödinger equation.
We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.
We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to …
The global attraction is proved for the nonlinear 3D Klein-Gordon equation with a nonlinearity concentrated at one point. Our main result is the convergence of each "finite energy solution" to the manifold of all solitary waves as $t\to\pm\infty$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap $[-m,m]$ and sa…
We consider the following generalized derivative nonlinear Schrödinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2σ}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $σ\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{ω,c}(t,x)=Φ_{ω,c}(x)e^{iωt+\frac{ic}2x-\frac i{2σ+2}\int_0^xΦ_{ω,c}(y)^{2σ}dy},$$ with $(ω,c)$ satisfying $ω>c^2/4$, or $ω=c^2/4$ and $c>0$. The stability theory in the frequency region $ω>c^2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $ω=c^2/4$ …
In this paper we study the behavior of solutions of a nonlinear Schroedinger equation in presence of an external potential, which is allowed to be singular at one point. We show that the solution behaves like a solitary wave for long time even if we start from a unstable solitary wave, and its dynamics coincide with that of a classical particle evolving according to a natural effective Hamiltonian.
Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrodinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrodinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrodinger equations in two and three dimensional domains, we prove the existence of least energy solutions and study the…
We study the existence of solitary-wave solutions and some of their properties for a general multicomponent long-wave-short-wave interaction system. The system considered here describes the nonlinear interaction of multiple short waves with a long-wave, and is of interest in plasma physics, nonlinear optics, and fluid dynamics.
The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection and method of majorants.
We consider the nonlinear Klein-Gordon equation in $\R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy est…
This paper concerns with the existence of solitons, namely stable solitary waves in the nonlinear beam equation (NBE) with a suitable nonlinearity. An equation of this type has been introduced by P.J. McKenna and W. Walter as a model of a suspension bridge. We prove both the existence of solitary waves for a large class of nonlinearities and their stability. As far as we know this is the first result about stability of solitary waves in NBE.
In this paper, we show the orbital stability of solitons arising in the cubic derivative nonlinear Schrodinger equations. We consider the zero mass case that is not covered by earlier works [8, 3]. As this case enjoys L^2 scaling invariance, we expect the orbital stability in the sense up to scaling symmetry, in addition to spatial and phase translations. For the proof, we are based on the variational argument and extend a similar argument in [21]. Moreover, we also show a self-similar type blow up criteria of solutions with the critical mass 4π.
A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational…
New exact solutions for the heat equation with a polynomial non-linearity and for the Fisher equation are found. An extended class of non-linear heat equations admitting solitary wave solutions is found. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.
We consider the nonlinear magnetic Schrödinger equation for $ u: \mathbb{R}^3 \times \mathbb{R} \to \mathbb{C} $, \[ iu_t = (i \nabla + A)^2 u + V u + g(u), u(x,0) = u_0(x),\] where $ A :\mathbb{R}^3 \to \mathbb{R}^3 $ is the magnetic potential, $ V : \mathbb{R}^3 \to \mathbb{R} $ is the electric potential, and $ g = \pm | u |^2 u $ is the nonlinear term. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in $ H^1 $, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as $t \to …
For the nonlinear Dirac equation in (1+1)D with scalar self-interaction (Gross--Neveu model), with quintic and higher order nonlinearities (and within certain range of the parameters), we prove that solitary wave solutions are asymptotically stable in the "even" subspace of perturbations (to ignore translations and eigenvalues $\pm 2ωi$). The asymptotic stability is proved for initial data in $H^1$. The approach is based on the spectral information about the linearization at solitary waves which we justify by numerical simulations. For the proof, we develop the spectral theory for the lineariz…
We consider the focusing $L^2$-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solution of NLS blowing up at several blow-up point together with a topological argument…
