[Paper Review] Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

Xiao Liu, Gideon Simpson+1 more|arXiv (Cornell University)|Jun 15, 2012

Advanced Mathematical Physics Problems31 references4 citations

TL;DR

This paper investigates the orbital stability of solitary wave solutions for a generalized derivative nonlinear Schrödinger equation (gDNLS) with power-type nonlinearity $\sigma > 0$. Using spectral analysis and numerical simulations, it establishes that stability depends critically on $\sigma$ and the soliton velocity $c$: solitary waves are unstable for $\sigma \geq 2$, stable for $\sigma < 1$, and exhibit a velocity-dependent stability threshold in the range $1 < \sigma < 2$, where slow solitons ($c < 2z_0\sqrt{\omega}$) are stable and fast solitons are unstable.

ABSTRACT

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.

Motivation & Objective

• To analyze the orbital stability of solitary wave solutions for a generalized derivative nonlinear Schrödinger equation (gDNLS) with power nonlinearity $\sigma > 0$.
• To determine how the strength of nonlinearity ($\sigma$) and soliton parameters ($\omega$, $c$) affect stability.
• To extend prior results on the standard DNLS ($\sigma = 1$) to a broader class of nonlinearities.
• To provide numerical evidence for a velocity-dependent stability threshold in the $1 < \sigma < 2$ regime.
• To establish conditions under which solitary waves are orbitally stable or unstable, based on spectral and variational analysis.

Proposed method

• The study considers the gDNLS equation $i\partial_t\psi + \partial_x^2\psi + i|\psi|^{2\sigma}\partial_x\psi = 0$, which admits two-parameter solitary wave solutions.
• Solitary wave solutions are constructed in the form $\psi_{\omega,c}(x,t) = \phi_{\omega,c}(x - ct)e^{i\omega t}$, with $\phi_{\omega,c}$ solving a nonlinear ODE.
• Orbital stability is analyzed via the variational structure of the Hamiltonian and conserved quantities (mass $Q$, momentum $P$), using their derivatives with respect to parameters $\omega$ and $c$.
• The stability criterion relies on the sign of the second variation of the action functional, derived from the spectral properties of the linearized operator around the solitary wave.
• Analytical expressions for $\partial_c Q$, $\partial_\omega Q$, $\partial_c P$, and $\partial_\omega P$ are derived using asymptotic expansions and integral identities involving hyperbolic functions.
• Numerical simulations are used to verify the stability threshold in the $1 < \sigma < 2$ range, identifying a critical velocity $c = 2z_0\sqrt{\omega}$ that separates stable and unstable regimes.

Experimental results

Research questions

• RQ1 For which values of the nonlinearity power $\sigma$ are solitary wave solutions of the generalized DNLS equation orbitally stable or unstable?
• RQ2 How does the soliton velocity $c$ influence the stability of solitary waves when $\sigma \in (1,2)$?
• RQ3 What is the critical velocity threshold $c_0(\sigma)$ that separates stable from unstable solitary waves in the $1 < \sigma < 2$ regime?
• RQ4 Why are solitary waves with $\sigma < 1$ always orbitally stable, regardless of velocity?
• RQ5 How does the stability behavior of the gDNLS equation differ from the classical DNLS equation ($\sigma = 1$)?

Key findings

• For $\sigma \geq 2$, all solitary wave solutions $\psi_{\omega,c}$ are orbitally unstable, regardless of $c$ or $\omega$, due to the strong nonlinearity destabilizing the wave profile.
• For $\sigma < 1$, all solitary wave solutions are orbitally stable, as confirmed by variational analysis and the positivity of the second variation of the action functional.
• In the range $1 < \sigma < 2$, a velocity-dependent stability threshold exists: solitary waves with $c < 2z_0\sqrt{\omega}$ are orbitally stable, while those with $c > 2z_0\sqrt{\omega}$ are unstable, where $z_0 \in (-1,1)$ is a $\sigma$-dependent constant.
• Numerical simulations confirm the existence of a critical velocity $c_0(\sigma)$ that separates stable and unstable regimes in the $1 < \sigma < 2$ case, with $z_0(\sigma)$ decreasing as $\sigma$ increases toward 2.
• The stability analysis relies on the derivatives of mass $Q$ and momentum $P$ with respect to $\omega$ and $c$, which are expressed in terms of integrals involving hyperbolic functions and shown to be proportional to $\tilde{\kappa}$, a scaling factor depending on $\sigma$, $\omega$, and $c$.
• The paper provides explicit analytical expressions for $\partial_c Q$, $\partial_\omega Q$, $\partial_c P$, and $\partial_\omega P$ in terms of $\alpha_n = \int_0^\infty h^{-1/\sigma - n} dx$, enabling the derivation of the stability threshold.