Anatoly G. Nikitin, Tetyana A. Barannyk
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.
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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.
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…
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 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…
Solitary waves of nonlinear Dirac, Maxwell-Dirac and Klein-Gordon-Dirac equations are considered. We prove that the energy-momentum relation for solitary waves coincides with the Einstein energy-momentum relation for point particles.
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…
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…
We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunctions. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A solitary wave which has a non-vanishing angular momentum is called vortex. We know (at least) three mechanisms which might produce solitary waves and vortices: 1) Complete integrability, (e.g. Kortewg-de Vries equation) 2) Topological constraints, (e.g. Sine-Gordon equation); 3) Ratio energy/charge: (e.g. the nonlinear Klein-Gordon equation). The third type of solitary waves or solitons will be called hylomorphic. This class includes the…
Solution of the nonlinear Klein-Gordon equation perturbed by small external force is investigated. The frequency of perturbation varies slowly and passes through a resonance. The resonance generates a solitary packets of waves. Full asymptotic description of this process is presented.
Properties of solitary waves in pre-compressed Hertzian chains of particles are studied in the long wavelength limit using a well-known continuum model. Several main results are obtained by parameterizing the solitary waves in terms of their wave speed and their asymptotic amplitude. First, the asymptotic amplitude is shown to be directly related to the continuum sound speed, and the ratio of asymptotic amplitude to peak amplitude is shown to describe the degree of dynamical nonlinearity in the underlying discrete system. Second, an algebraic relation is derived that determines the dynamical n…
Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity
本文研究具有集中Soler型非线性的Dirac方程孤立波的光谱稳定性，证明仅打破SU(1,1)对称性而不破坏宇称对称性的扰动仍保持稳定性，而同时打破两种对称性的扰动会通过嵌入特征值分岔导致弱相对论孤立波失稳。
We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i \pt_t u = \sqrt{-Δ} \, u - |u|^{p-1} u, \quad \mbox{with $(t,x) \in \R \times \R^d$} $$ with exponents $1 < p < \infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d \geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{iωt} Q_v(x-vt) $$ with frequency $ω\in \R$, velocity $v \in \R^d$, and some finite-energy profile $Q_v \in H^{1/2}(\R^d)$, $Q_v \not \equiv 0$. We prove that traveling solitary waves for speeds $|v| \geq 1$ do not exist. Furthermore, we generalize the non-existence result to the sq…
In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schrödinger system \[ i\partial_t u_{j}+\partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj} |u_k|^{p}\right)|u_j|^{p-2}u_j = 0, \ j=1,2,3, \] where $u_j$ are complex-valued functions of $(x,t)\in \mathbb{R}^{2}$ and $a_{kj}$ are positive constants satisfying $a_{kj}=a_{jk}$ (symmetric attractive case). Our approach improves many of the previous known results. In all methods used previously to study solitary waves, which we are aware of, the variational problem has c…
The $2$D Benney-Luke equation is an isotropic model which describes long water waves of small amplitude in $3$D whereas the KP-II equation is a unidirectional model for long waves with slow variation in the transverse direction. In the case where the surface tension is weak or negligible, linearly stability of small line solitary waves of the $2$D Benney-Luke equation was proved by Mizumachi and Shimabukuro [Nonlinearity, 30 (2017), 3419--3465]. In this paper, we prove nonlinear stability of the line solitary waves by adopting the argument by Mizumachi ([Mem. Amer. Math. Soc. no. 1125], [Proc.…
Solitary wave and soliton solutions of nonlinear equations are well known for physicists. A soliton is a solitary wave with some outstanding features which make it reasonable to be studied seriously in nonlinear systems. In fact most of the nonlinear systems which have solitary wave solutions, has no soliton solutions. To realize a solitary wave as a soliton, we must prepare some initial conditions to collide two or more solitary wave solutions. In fact it is not possible to prepare such initial conditions for any nonlinear system with solitary wave solutions. In this paper we study the condit…
The solitary waves of massive (1+1)-dimensional nonlinear S^N-sigma models are unveiled. It is shown that the solitary waves in these systems are in one-to-one correspondence with the separatrix trajectories in the repulsive N-dimensional Neumann mechanical problem. There are topological (heteroclinic trajectories) and non-topological (homoclinic trajectories) kinks. The stability of some embedded sine-Gordon kinks is discussed by means of the direct estimation of the spectra of the second-order fluctuation operators around them, whereas the instability of other topological and non-topological…
We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{k+1} ({\bar Ψ} Ψ)^{k+1}$, as well as a vector-vector self interaction $\frac{g^2}{k+1} ({\bar Ψ} γ_μΨ\bPsi γ^μΨ)^{\frac{1}{2}(k+1)}$. We find the exact analytic form for solitary waves for arbitrary $k$ and find that they are a generalization of the exact solutions for the nonlinear Schrödinger equation (NLSE) and reduce to these solutions in a well defined nonrelativistic limit. We perform the nonrelativistic reduction and find the $1/2m$ correction to the NLSE, valid when $|ω-m…
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…
This paper concentrates on a (1+1)-dimensional nonlinear Dirac (NLD) equation with a general self-interaction, being a linear combination of the scalar, pseudoscalar, vector and axial vector self-interactions to the power of the integer $k+1$. The solitary wave solutions to the NLD equation are analytically derived, and the upper bounds of the hump number in the charge, energy and momentum densities for the solitary waves are proved in theory. The results show that: (1) for a given integer $k$, the hump number in the charge density is not bigger than $4$, while that in the energy density is no…
For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly-decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions: The first one is in the realm of the discrete nonlinear Schrodinger equation, where the linear operator of the Schrodinger type is considered in the presence of a Kerr focusing or defocusing nonlinearity and the embedded linear mode is continued into the nonlinear regime as a discrete solitary wave. The second case is the Klein-Gordon setting, where the pres…
We study statically homogeneous Bose-Einstein condensates with spatially inhomogeneous interactions and outline an experimental realization of compensating linear and nonlinear potentials that can yield constant-density solutions. We illustrate how the presence of a step in the nonlinearity coefficient can only be revealed dynamically and consider, in particular, how to reveal it by exploiting the inhomogeneity of the sound speed with a defect-dragging experiment. We conduct computational experiments and observe the spontaneous emergence of dark solitary waves. We use effective-potential theor…
The solitary wave problem at the free surface of a two-dimensional, infinitely-deep and irrotational flow of water, under the influence of gravity, is formulated as a nonlinear pseudodifferential equation. A Pohozaev identity is used to show that it admits no solutions which asymptotically vanish faster than linearly.
We consider a U(1)-invariant nonlinear Klein-Gordon equation in dimension one or larger, self-interacting via the mean field mechanism. We analyze the long-time asymptotics of finite energy solutions and prove that, under certain generic assumptions, each solution converges (as time goes to infinity) to the two-dimensional set of all ``nonlinear eigenfunctions'' of the form $ϕ(x)e\sp{-iωt}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.
We study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature and then we can obtain the transformation for the 1+1 dimensional Dirac equation directly from the metric. Depending on the curvature, the soliton profile either narrows or expands. Our results may be applicable to yet-to-be-synthesized curved quasi-one dimensional Bose condensates.
Solitary wave-mean flow interaction in strongly nonlineardispersive shallow water waves
本文利用 Whitham 调制理论和 Riemann 不变量解析研究了 SGN 方程中孤立波与平均流的相互作用，揭示了孤立波的捕获与透射机制，并证明了基于 DSW 拟合的近似方法具有二阶精度。
Solitary waves of relativistic invariant nonlinear wave equation with symmetry group U(1) are considered. We prove that the energy-momentum relation for spherically symmetric solitary waves coincides with the Einstein energy-momentum relation for point particles.
The stability of topological solitary waves and pulses in one-dimensional nonlinear Klein-Gordon systems is revisited. The linearized equation describing small deviations around the static solution leads to a Sturm-Liouville problem, which is solved in a systematic way for the $-l\,(l+1)\,\sech^2(x)$-potential, showing the orthogonality and completeness relations fulfilled by the set of its solutions for all values $l\in\mathbb{N}$. This approach allows to determine the linear stability of kinks and pulses of certain nonlinear Klein-Gordon equations. Two families of novel nonlinear Klein-Gordo…
Generalized PT-symmetric nonlinear Dirac equation: exact solitary waves solutions, stability and conservation laws
本文研究了PT对称非线性狄拉克方程的孤立波解，探讨了其稳定性及守恒律，发现非线性指数不影响PT临界点，并展示了守恒量与动量的关系。
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