Dissipative soliton dynamics of the Landau–Lifshitz–Gilbert equation

Abstract

We study ferromagnetic dissipative systems described by the isotropic LLG equation, from the standpoint of their spatially localized dynamical excitations. In particular, we focus on dissipative soliton solutions of a nonlocal NLS equation to which the LLG equation is transformed and use Melnikov’s theory to prove the existence of these solutions for sufficiently small dissipation. Next, we employ pseudospectral and PINN (physics-informed neural network) numerical techniques of machine learning to demonstrate the validity of our analytic results. Such localized structures have been detected experimentally in magnetic systems and observed in nano-oscillators, while dissipative magnetic droplet solitons have also been found theoretically and experimentally.

Appendix: Homoclinic Melnikov theory of integro-differential equations

Here, we briefly review the theory of a more general class of perturbed nonlocal dynamical systems [24], which we used in Sec. 2 to study the existence and stability of bright solitons in the nonlocal NLS (2.2).

We start by considering a perturbed $n$-dimensional dynamical system of the integro-differential form

$$ \dot x=f(x)+\epsilon [\kern1pt\mathcal N\kern1.5pt] (x,t;\mu),\qquad x\in\mathbb{R}^{n},$$

(A.1)

where $0\le|\epsilon|\ll 1$, $[\kern1pt\mathcal N\kern1.5pt]$ are periodic functions in $t$, and $\mu\in\mathbb{R}$ is the set of external parameters of the problem. The nonlocal part $\mathcal N$ is quite general and includes different classes of terms that arise frequently in applications.

We assume that the unperturbed system (A.1) (corresponding to $\epsilon=0$) has a hyperbolic singular point $x_0$ and a homoclinic orbit $\gamma$ to $x_0$. The linear variational equation of unperturbed system (A.1) along the homoclinic orbit $\gamma$, written in the form

$$ \dot z=A(t)z,\qquad A(t)=\mathrm Df(\gamma(t)),$$

(A.2)

is known to have dichotomies on $\alpha\le t\le\infty$ and $-\infty\le t\le\alpha$ for $\alpha\in\mathbb{R}$. Clearly, when $A(t)=A$ is constant, system (A.2) has an exponential dichotomy on the infinite interval if and only if $\operatorname{spec}A=\{\lambda\in\mathbb{C}\colon\operatorname{Re}\lambda\neq 0\}$, whereas when $A(t)=A(t+T)$, Eq. (A.2) has an exponential dichotomy if and only if the Floquet multipliers lie outside the unit circle [25].

We now let $W^{\mathrm s}_0$ and $W^{\mathrm u}_0$ denote the respective stable and unstable manifolds of the singular point $x_0$, and define the following projections for a given $\alpha\in\mathbb{R}$:

$$P(a)\colon T_{\gamma(\alpha)}\mathbb{R}^{n}\to T_{\gamma(\alpha)}W^{\mathrm s}_0,\qquad Q(a)\colon T_{\gamma(\alpha)}\mathbb{R}^{n}\to T_{\gamma(\alpha)}W^{\mathrm u}_0$$

satisfying,

$$(I-P(\alpha))T_{\gamma(\alpha)}\mathbb{R}^{n}=T^{\perp}_{\gamma(\alpha)}W^{\mathrm s}_0,\qquad (I-Q(\alpha))T_{\gamma(\alpha)}\mathbb{R}^{n}=T^{\perp}_{\gamma(\alpha)}W^{\mathrm u}_0.$$

Here, $T_{\gamma(\alpha)}\mathbb{R}^{n}$ and $T^{\perp}_{\gamma(\alpha)}W^{\mathrm s}_0$ are the tangent spaces to $\mathbb{R}^{n}$ and to $W^{\mathrm s}_0$ at $\gamma(\alpha)$. System (A.2) has an exponential dichotomy on $\alpha\le t\le\infty$, such that $\Phi(t,\alpha)z_0\to 0$ exponentially as $t\to\infty$ if $z_0\in T_{\gamma(\alpha)}W^{\mathrm u}_0$, where $\Phi(t,s)$ is the Floquet transition matrix of (A.2).

We now define the perturbed stable invariant manifold of the fixed point $x_\epsilon$ as

$$ W^{\mathrm s}_{\mathrm{loc}}(x_\epsilon)= \bigcup_{\alpha\in\mathbb{R}}\bigl\{\gamma(\alpha)+\epsilon\kern1pt G^{\kern1pt\mathrm s}_2(\alpha,\eta^{\mathrm s};\epsilon;\mu)\bigr\},$$

(A.3)

where

$$G^{\kern1pt\mathrm s}_2(\alpha,\eta^{\mathrm s};\epsilon;\mu)= \eta^{\mathrm s}+(I-P(\alpha))\int^{\alpha}_{\infty}\Phi(\alpha,\tau) [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)\,d\tau$$

and $\eta^{\mathrm s}\in T_{\gamma(\alpha)}W^{\mathrm s}_0/T_{\gamma(\alpha)}\gamma(\alpha)$, $|\eta^{\mathrm s}|\ll 1$. Similarly, for the perturbed unstable manifold, we have

$$ W^{\mathrm u}_{\mathrm{loc}}(x_\epsilon)= \bigcup_{\alpha\in\mathbb{R}}\bigl\{\gamma(\alpha)+\epsilon\kern1pt G^{\mathrm u}_2(\alpha,\eta^{\mathrm u};\epsilon;\mu)\bigr\}.$$

(A.4)

Here,

$$G^{\mathrm u}_2(\alpha,\eta^{\mathrm u};\epsilon;\mu)= \eta^{\mathrm u}+(I-Q(\alpha))\int^{\alpha}_{-\infty}\Phi(\alpha,\tau) [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)\,d\tau$$

where $\eta^{\mathrm u}\in T_{\gamma(\alpha)}W^{\mathrm u}_0/T_{\gamma(\alpha)}\gamma(\alpha)$, $|\eta^{\mathrm u}|\ll 1$. Choosing

$$ x(t)=\gamma(t+\alpha)+\epsilon z_2(t+\alpha),\qquad \alpha\in\mathbb{R}$$

(A.5)

as a perturbation of the homoclinic orbit of (A.1), we obtain the variational equations

$$ \dot z_2=A(t)z_2+ [\kern1pt\mathcal N\kern1.5pt] (\gamma(t), t-\alpha;\mu)=h_2(t,x,z,\epsilon),$$

(A.6)

where

$$ h_2(t,x,z,\epsilon)= \frac{1}\epsilon\bigl\{f(\gamma(t)+\epsilon z_2)-f(\gamma(t))z_2\big\}+ [\kern1pt\mathcal N\kern1.5pt] (\gamma(t)+\epsilon z_2)- [\kern1pt\mathcal N\kern1.5pt] (\gamma(t)).$$

(A.7)

Let $z_2(t;\alpha,z_0)$ be a solution of system (A.6) with $z_2(0;\alpha,z_0)=z_{0,2}$. Following [20], it is straightforward to conclude that $z_2(t;\alpha,z_0)$ is bounded for $\alpha\le t<\infty$ if and only if it satisfies

$$\begin{aligned} \, z_2(t;\alpha,z_0)={}& \Phi(t,\alpha) \biggl[P(\alpha)z_{0,2}+P(\alpha)\int^t_{\alpha}\Phi(\alpha,\tau)\{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z_2;\epsilon)\}\,d\tau+{} \nonumber\\ &+(I-P(\alpha))\int^t_{\infty}\Phi(\alpha,\tau)\{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z_2;\epsilon)\}\,d\tau\biggr]. \end{aligned}$$

(A.8)

Letting $\eta^{\mathrm s}_2=P(\alpha)z_{0,2}$, we can prove, using the contraction mapping theorem, that system (A.8) has the unique solution

$$z^{\mathrm s}_2=z_2(t;\alpha, z_{0,2}(\eta^{\mathrm s}_2)),\qquad |\eta^{\mathrm s}_1|\ll 1,\quad |\eta^{\mathrm s}_2|\ll 1.$$

Setting now $t=\alpha$ in (A.8), we obtain

$$ z_{0,2}(\eta^{\mathrm s}_2)=\eta^{\mathrm s}_2+ (I-P(\alpha))\int^{\alpha}_{\infty}\Phi(\alpha,\tau)\{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z_2;\epsilon)\}\,d\tau,$$

(A.9)

whence

$$ x(0)=\gamma(\alpha)+\epsilon \biggl[\eta^{\mathrm s}_2+(I-P(\alpha))\int^{\alpha}_{\infty}\Phi(\alpha,\tau) \{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z_2;\epsilon;\mu)\}\,d\tau\biggr].$$

(A.10)

It follows that we can obtain the perturbed local stable and unstable manifolds of the Poincaré map of the perturbed dynamical system (A.1), $W^{\mathrm s}_{\mathrm{loc}}(x_\epsilon)$ and $W^{\mathrm u}_{\mathrm{loc}}(x_\epsilon)$, as the graphs on the unperturbed stable and unstable manifolds $W^{\mathrm s}_0$ and $W^{\mathrm u}_0$. The functions $G^{\mathrm s}(\alpha,\eta^{\mathrm s},\epsilon;\mu)$ and $G^{\mathrm u}(\alpha,\eta^{\mathrm u},\epsilon;\mu)$ decompose as (see [20])

$$\begin{aligned} \, &G^{\mathrm s}(\alpha,\eta^{\mathrm s},\epsilon;\mu)= (\alpha,\nu,\eta^{\mathrm s}_2,m^{\mathrm s}_2(\alpha,\nu,\eta^{\mathrm s}_1,\eta^{\mathrm s}_2,\epsilon;\mu)), \\ &G^{\mathrm u}(\alpha,\eta^{\mathrm u},\epsilon;\mu)= (\alpha,\nu,\eta^{\mathrm u}_2,m^{\mathrm u}_2(\alpha,\nu,\eta^{\mathrm u}_1,\eta^{\mathrm u}_2,\epsilon;\mu)), \end{aligned}$$

where $(\alpha,\nu)\in\operatorname{Range}P(\alpha)\cap\operatorname{Range}Q(\alpha)$ and $\eta^{\mathrm s,\mathrm u}=(\nu,\eta^{\mathrm s,\mathrm u}_2)$ with

$$\eta^{\mathrm s}_2\in\operatorname{Range}P(\alpha)\cap\operatorname{Range}(I-Q(\alpha)),\qquad \eta^{\mathrm u}_2\in\operatorname{Range}(I-P(\alpha))\cap\operatorname{Range}Q(\alpha)$$

(see [20]).

To measure the distance between the stable and unstable manifolds of the perturbed Poincaré map, it suffices to measure the separation of these manifolds in the subspace $\operatorname{Range}(I-P(\alpha))\cap\operatorname{Range}(I-Q(\alpha))$. Let $\mathcal D$ denote the distance between $W^{\mathrm s}_{\mathrm{loc}}(x_{\epsilon,\mu})$ and $W^{\mathrm u}_{\mathrm{loc}}(x_{\epsilon,\mu})$:

$$ \mathcal D=m^{\mathrm u}_2(\alpha,\nu,\eta^{\mathrm u}_2,\epsilon;\mu)-m^{\mathrm s}_2(\alpha,\nu,\eta^{\mathrm s}_2,\epsilon;\mu).$$

(A.11)

We consider the problem adjoint to (A.2),

$$ \dot\phi+(\mathrm Df(\gamma(t)))^*\phi=0$$

(A.12)

and let $\{\phi_i\}$, $i=1,\ldots m$, be a set of linearly independent bounded solutions on $\mathbb{R}$. Then

$$\operatorname{Range}(I-P^*(\alpha))\cap\operatorname{Range}(I-Q^*(\alpha))=\operatorname{span}\{\phi_i^*(\alpha)\}^{m}_{i=1}.$$

In terms of $\phi_1(\alpha),\ldots,\phi_m(\alpha)$, the manifold separation $\mathcal D$ has the coordinate expression

$$\begin{aligned} \, \phi^*_i(\alpha)\mathcal D={}&\phi^*_i(\alpha)(m^{\mathrm u}_2-m^{\mathrm s}_2) \biggl[\phi^*_i(\alpha)(I-Q(\alpha)) \int^{\alpha}_{-\infty}\!\!\Phi(\alpha,\tau) \{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z^{\mathrm u}_1,z^{\mathrm u}_2;\epsilon)\}\,d\tau\biggr]-{} \\ &-\biggl[\phi^*_i(\alpha)(I-P(\alpha)) \int^{\alpha}_{\infty}\!\!\Phi(\alpha,\tau) \{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z^{\mathrm s}_1,z^{\mathrm s}_2;\epsilon)\}\,d\tau\biggr]= \\ ={}&\biggl[\,\int^{\alpha}_{-\infty}\!\!\phi^*_i(\tau) \{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z^{\mathrm u}_1,z^{\mathrm u}_2;\epsilon)\}\,d\tau\biggr]-{} \\ &-\biggl[\,\int^{\alpha}_{\infty}\!\!\phi^*_i(\tau) \{ [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)+h_2(\tau,\alpha,z^{\mathrm s}_1,z^{\mathrm s}_2;\epsilon)\}\,d\tau\biggr]= \\ ={}&\int^{\infty}_{-\infty}\!\!\phi^*(t) [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)\,dt+{} \\ &+\biggl[\,\int^{0}_{-\infty}\!\!\phi^*_i(\tau) h_2(\tau,\alpha,z^{\mathrm u}_1,z^{\mathrm u}_2;\epsilon)\,d\tau- \int^{0}_{\infty}\!\!\phi^*_i(\tau)h_2(\tau,\alpha,z^{\mathrm s}_1,z^{\mathrm s}_2;\epsilon)\,d\tau\biggr] \end{aligned}$$

for $i=1,2,\ldots,m$.

We define $\mathcal D=\epsilon M_2+O(\epsilon^2)$, where the components of the Melnikov vector $M_2$ are given by

$$ M_{2,i}(\alpha,\nu)=\int^{\infty}_{-\infty}\phi^*_i(t) [\kern1pt\mathcal N\kern1.5pt] (\gamma(\tau),\tau-\alpha;\mu)\,dt,\qquad i=1,2,$$

(A.13)

where $z_1$ is the solution of the first variational problem $\dot z_1=\mathrm Df(\gamma(t))z_1$.

Thus, the conditions for the existence of transverse intersections between the stable and unstable manifolds of a Poincaré map associated with the system of integro-differential equations (A.1) can be summarized by the following theorem [24].

Theorem.

If there exist $\alpha$, $\nu=(\nu_1,\ldots,\nu_{m-1})$, and $\epsilon$ such that the Melnikov vector $M_1(\alpha,\nu)=0$ and the $m$ vectors $\frac{\partial M_1}{\partial\alpha},\frac{\partial M_1}{\partial\nu_1},\ldots,\frac{\partial M_1}{\partial\nu_{m-1}}$ are all nonzero at the point $(\alpha,\nu)$, then the local stable and unstable manifolds of the Poincaré map associated with system (A.1) intersect transversally. Moreover, in the case where the Melnikov vector $M_1(\alpha,\nu)\equiv 0$, if there exist $\alpha$, $\nu=(\nu_1,\ldots,\nu_{m-1})$, and $\epsilon$ such that $M_2(\alpha,\nu)=0$ and the $m$ vectors $\frac{\partial M_2}{\partial\alpha},\frac{\partial M_2}{\partial\nu_1},\ldots,\frac{\partial M_2}{\partial\nu_{m-1}}$ are all nonzero at the point $(\alpha,\nu)$, then $W^{\mathrm s}_{\mathrm{loc}}(x_\epsilon)$ and $W^{\mathrm u}_{\mathrm{loc}}(x_\epsilon)$ intersect transversally.

We remark that the stability of bright solitons of the nonlocal NLS with cubic and quintic nonlocal terms was studied in [26] using a different approach to analyze their dynamical behavior.