Stability Of Solitary Waves Research Articles

Novel exact solutions to the perturbed Gerdjikov–Ivanov equation

This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four distinct exact and explicit solutions, along with the corresponding first integrals. Explanations and presentations of solutions are given in a logical manner. We derive an analytical expression for the instability gain and examine its key features using linear stability analysis. Finally, we compare the correctness of the analytical and numerical solutions. We demonstrate the robustness and stability of solitary waves through numerical simulations.

Existence and stability of solitary waves to the rotation–Camassa–Holm equation

In this paper, we investigate the existence and stability of solitary waves to the rotation–Camassa–Holm equation which can be considered as a model in the shallow water for the long‐crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. We prove the existence of solitary waves by performing a phase plane analysis. Moreover, utilizing the approach proposed by Grillakis–Shatah–Strauss, we prove the stability of solitary waves.

Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes

We consider perturbations of the one-dimensional cubic Schrödinger equation, under the form i∂tψ+∂x2ψ+|ψ|2ψ−g(|ψ|2)ψ=0. Under hypotheses on the function g that can be easily verified in some cases, we show that the linearized problem around a solitary wave does not have internal mode (nor resonance) and we prove the asymptotic stability of these solitary waves, for small frequencies.

On the stability of solitary waves in the NLS system of the third-harmonic generation

On the stability of solitary waves in the NLS system of the third-harmonic generation

Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation

Abstract: We consider the quadratic Zakharov-Kuznetsov equation $$\partial_t u + \partial_x \Delta u + \partial_x u^2=0$$ on $\Bbb{R}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is the ground state solution to $-Q+\Delta Q+Q^2=0$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $Q$ in the energy space, evolves to a solution that, as $t\to\infty$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $x>\delta t-\tan\theta\sqrt{y^2+z^2}$ for $0\leq\theta\leq{\pi\over 3}-\delta$.

Local Smoothing Estimates for Schrödinger Equations on Hyperbolic Space

We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space H d \mathbb {H}^d , d ≥ 2 d \geq 2 . In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood–Paley machinery on H d \mathbb {H}^d based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on H d \mathbb {H}^{d} . Specifically, some of the estimates established in this paper play a crucial role in the authors’ proof of the nonlinear asymptotic stability of harmonic maps under the Schrödinger maps evolution on the hyperbolic plane; see Lawrie, Lührmann, Oh, and Shahshahani, 2023. As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-dependent perturbations, including polynomially decaying second order ones, and small lower order nonsymmetric perturbations.

Orbital Stability of Solitary Wave for Eckhaus–Kundu Equation

In this paper, the orbital stability of solitary wave for Eckhaus–Kundu equation is studied. Since the equation we studied is difficult to be expressed as a standard Hamiltonian system, the Grillakis–Shatah–Strauss theory about the orbital stability of soliton solutions for nonlinear Hamiltonian systems cannot be directly applied. By constructing three new conserved quantities and using special techniques and detailed spectral analysis, the above difficulty is overcome, then we obtain the conclusion that the solitary wave of Eckhaus–Kundu equation is orbitally stable.

Orbital stability of solitary waves for a two-component Novikov system

Orbital stability of solitary waves for a two-component Novikov system

The wave stability of solitary waves over a bump for the full Euler equations

The wave stability of solitary waves over a bump for the full Euler equations

Bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equations

We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger (CNLS) equations on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a fundamental solitary wave. We establish criteria under which the fundamental solitary wave undergoes a pitchfork bifurcation, and utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral and/or orbital stability of the bifurcated solitary waves as well as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. We apply our theory to a cubic nonlinearity case and give numerical evidences for the theoretical results.

The long-time behavior of solitary waves for the weakly damped KdV equation

In this paper, we first introduce the long-time behavior stability of solitary waves for the weakly damped Korteweg–de Vries equation. More concretely, solutions of the dissipative system with the initial values near a c_{0}-speed solitary wave, are approximated by a long curve on the family of solitary waves with the time-varying speed |c(t)-c_{0}| being small, in the long-time period (i.e., 0leqslant tleqslant O(frac{1}{epsilon ^{tau}})). Meanwhile, the approximation difference in a suitably weighted space H^{1}_{a}(mathbb{R}) is of the order of the damping coefficient and of some kind of exponential weight form. As a comparison, we also study the long-time behavior stability, i.e., for 0leqslant t<+infty , the solutions are approximated by a long curve on the family of solitary waves with the exponential decay speed c(t)= c_{0}e^{-beta t} (0<beta leqslant 1), when the initial values are near a c_{0}-speed solitary wave. However, here, the approximation difference merely defined in H^{1}(mathbb{R}) depends on the damping coefficient ϵ and the exponential decay coefficient β.

Asymptotic stability of solitary waves for the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ 1d $&lt;/tex-math&gt;&lt;/inline-formula&gt; NLS with an attractive delta potential

We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result [35] beyond the regime of small solitary waves and extend the results of [19,29] from orbital to asymptotic stability for a suitable family of solitary waves.

Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode

For the Schr\"odinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove the asymptotic stability of solitary waves for a large range of admissible frequencies. For this model, the linearized problem around the solitary waves does not have internal mode nor resonance.

Stability of solitary waves in nonlinear Klein–Gordon equations

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 -potential, showing the orthogonality and completeness relations fulfilled by the set of its solutions for all values . This approach enables the linear stability of kinks and pulses of certain nonlinear Klein–Gordon equations to be determined. The inverse problem, which starts from Sturm–Liouville problem and obtains nonlinear Klein–Gordon potentials, is also revisited and solved in a direct way. The exact solutions (kinks and pulses) for these potentials are calculated, even when the nonlinear potential is not explicitly known. The kinks are found to be stable, whereas the pulses are unstable. The stability of the pulses is achieved by introducing certain spatial inhomogeneities.

Orbital stability of solitary waves on a ferrofluid jet

We prove the orbital stability of small-amplitude axisymmetric solitary waves on the surface of an incompressible, inviscid ferrofluid jet. The ferrofluid surrounds a current-carrying rod and is subject to the azimuthal magnetic field generated by the rod. We show that under appropriate assumptions on the magnitude of the magnetic intensity in the ferrofluid, both the trivial flow and the solitary waves with strong surface tension are conditionally orbitally stable, while the conditional orbital stability of solitary waves with near-critical surface tension can be deduced from properties of the corresponding dispersive PDE model equation. The arguments are based on the recent orbital stability results for internal waves by Chen and Walsh (2022) and an improved version of the Grillakis–Shatah–Strauss method introduced by Varholm et al. (2020).

Klein–Gordon equation with mean field interaction. Orbital and asymptotic stability of solitary waves * *Research supported by the Austrian Science Fund (FWF) under Grant No. P 34177.

We investigate orbital and asymptotic stability of solitary wave solutions to the U(1)-invariant nonlinear Klein–Gordon equation with mean field interaction.

ORBITAL STABILITY OF SOLITARY WAVES FOR THE NONLINEAR SCHRÖDINGER-KDV SYSTEM

<abstract> This paper investigates the stability of solitary waves for the nonlinear Schrödinger-KdV system. We establish the existence and orbital stability of solitary waves solutions by applying the abstract results and detailed spectral analysis, and this result improves the previous one by Chen (1999). </abstract>

Linear Stability of Solitary Waves for the Isothermal Euler–Poisson System

We study the asymptotic linear stability of a two-parameter family of solitary waves for the isothermal Euler–Poisson system. When the linearized equations about the solitary waves are considered, the associated eigenvalue problem in \(L^2\) space has a zero eigenvalue embedded in the neutral spectrum, i.e., there is no spectral gap. To resolve this issue, use is made of an exponentially weighted \(L^2\) norm so that the essential spectrum is strictly shifted into the left-half plane, and this is closely related to the fact that solitary waves exist in the super-ion-sonic regime. Furthermore, in a certain long-wavelength scaling, we show that the Evans function for the Euler–Poisson system converges to that for the Korteweg–de Vries (KdV) equation as an amplitude parameter tends to zero, from which we deduce that the origin is the only eigenvalue on its natural domain with algebraic multiplicity two. We also show that the solitary waves are spectrally stable in \(L^2\) space. Moreover, we discuss (in)stability of large amplitude solitary waves.

Numerical computations for bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equations

We numerically study solitary waves in the coupled nonlinear Schr\"odinger 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 its stability type at the saddle-node bifurcation point.

Solitary matter wave in spin-orbit-coupled Bose-Einstein condensates with helicoidal gauge potential.

We analytically and numerically study the different types of solitary wave in the two-component helicoidal spin-orbit coupled Bose-Einstein condensates (BECs). Adopting the multiscale perturbation method, we derive the analytical bright and dark solitary wave solutions of the system, and the stationary and moving bright (dark) solitary waves are obtained. The effects of spin-orbit coupling, the helicoidal gauge potential, the momentum, the Zeeman splitting, and the atomic interactions on the solitary wave types are discussed, and it is found that the coupling of these physical parameters can manipulate different types of solitary waves in the system. The results indicate that the helicoidal gauge potential breaks the symmetric properties of the energy band of the system and adjusts the energy band structure, thus further effecting the solitary wave properties, i.e., stationary or moving solitary wave, bright, or dark solitary wave. Correspondingly, the analytical predictions for exciting stationary or moving bright (dark) solitary wave in parameter space are obtained. In particular, the helicoidal gauge potential changes the solitary wave types drastically for the weak spin-orbit coupling, i.e., in the absence of the helicoidal gauge potential, only dark (bright) solitary wave solutions exist in the system with repulsive (attractive) atomic interaction; however, in the presence of the helicoidal gauge potential, both dark and bright solitary waves can exist in the system regardless of whether the atomic interaction is repulsive or attractive. In addition, we investigate the stability of solitary waves and obtain the stability regions of different types of solitary waves by applying the linear stability analysis. The dynamic evolution results of the solitary waves by the direct numerical simulation not only validate the linear stability analysis but also confirm the analytical prediction of the solitary waves. Finally, the collision effects between solitary waves are also presented by the numerical simulation. It is shown that the interactions between solitary waves in the system have both elastic and inelastic collisions, which are closely related to the position of solitary wave states in the linear energy band. Our results provide a potential way to adjust the types of solitary waves in BECs with helicoidal gauge potential.