Growth or decay of a coherent structure interacting with random waves

Solitary waves interacting with random Rayleigh-Jeans distributed waves of a nonintegrable and noncollapsing nonlinear Schrödinger equation are studied. Two opposing types of dynamics are identified: First, the random thermal waves can erode the solitary wave; second, this structure can grow as a result of this interaction. These two types of behavior depend on a dynamical property of the solitary wave (its angular frequency), and on a statistical property of the thermal waves (the chemical potential). These two quantities are equal at a saddle point of the entropy that marks a transition between the two types of dynamics: high-amplitude coherent structures whose frequency exceeds the chemical potential grow and smaller structures with a lower frequency decay. Either process leads to an increase of the wave entropy. We show this using a thermodynamic model of two coupled subsystems, one representing the solitary wave and one for the thermal waves. Numerical simulations verify our results.

Figure 1

Decay of a solitary wave interacting with thermal waves. Equation (1) is integrated over $2\times {10}^5$ time units. The initial state contains a solitary wave with a frequency ${\omega }_s=-0.1$ and an amplitude $|{\varphi }_m{|}^2\approx 0.238$. It is immersed in thermal waves with a chemical potential $\mu =-0.2$; the average amplitude is $\langle |\varphi {|}^2\rangle \approx 0.008$. The system size is $L=4096$ with periodic boundary conditions. $x=0$ in this plot is arbitrarily set in a way that the boundary $x=0$ or $x=4096$ is not crossed by the solitary wave in this particular simulation. (a) Time evolution (moving average over 2000 time units) of the spatial maximum of ${\left|\varphi \right|}^2$. The solitary wave decays from its initial amplitude ${\left|\varphi \right|}^2\approx 0.238$ below the amplitude of the background waves. (b) Initial amplitude profile ${\left|\varphi (x,t=0)\right|}^2$ at the starting point of the simulation; the solitary wave is located near $x=4000$. (c) Locations with ${\left|\varphi (x,t)\right|}^2\ge 0.16$ tracing the decaying solitary wave in space and time. (d) Amplitude profile $|\varphi (x,t=2\times {10}^5){|}^2$ at the end point of the simulation, the solitary wave is indistinguishable from the surrounding waves.

Figure 2

Growth of a solitary wave interacting with thermal waves. The thermal waves have a chemical potential $\mu =-0.05$, everything else corresponds to Fig. 1. The solitary wave is initially located at $x\approx 1000$. Its amplitude grows during time evolution, it is ${\left|\varphi \right|}^2\approx 0.5$ at $t=2\times {10}^5$.

Figure 3

(a) Potential $V\left(u\right)$ (4) for ${\omega }_s=-0.1$ and for ${\omega }_s=-3/16$. (b) Homoclinic orbits $u\left(x\right)$, $u_{xx}=-dV/du$ of the saddle point at $u_1=0$ for ${\omega }_s=-0.1$ and for ${\omega }_s\gtrsim -3/16$. The homoclinic orbit for ${\omega }_s=-0.1$ corresponds to a nontraveling solitary wave with a shape similar to the soliton of the focusing nonlinear Schrödinger equation. The homoclinic orbit for ${\omega }_s\gtrsim -3/16$ corresponds to a nontraveling solitary wave with a broad plateau at its maximum.

Figure 4

Frequency ${\omega }_s\left(A_s\right)=d{E}_s\left(A_s\right)/d{A}_s$ (6) of solitary waves. Inset: Lower boundary $E_s\left(A\right)$ (5) of the Hamiltonian, points on the line correspond to nontraveling solitary waves. Any other states (traveling solitary waves, spatially extended waves, etc.) correspond to points above this line. The Hamiltonian cannot have values in the shaded area below this line.

Figure 5

Thermodynamic model: The dynamical coupling of a solitary wave and thermal waves is reduced to a thermodynamic coupling that enables the transfer of energy and wave action between the two systems. The sketch shows a growth process of the solitary wave: Wave action is transferred from the thermal waves to the solitary wave, energy is transferred in the opposite direction. Flows of each conserved quantity in the opposite direction lead to a decreasing solitary wave. Which of the two processes occurs depends on whether the flow of energy or of wave action has a greater entropic effect, which in turn depends on the chemical potential $\mu$ of the thermal waves and the frequency ${\omega }_s$ of the solitary wave.

Figure 6

(a) Lower boundary $E_s\left(A_s\right)$ of the Hamiltonian corresponding to of the energy of solitary waves. (b) Wave entropy surface $S(A_w,E_w)$ of thermal waves for $\beta =10$, $\mu =-0.1$. The green line is an isentrope $S=\text{const},$ with $d{E}_w/d{A}_w=\mu$. The energy and wave action of the thermal waves can change by transfers of these quantities from or to growing or decaying solitary waves. This corresponds to a time evolution along the boundary curve from (a) projected onto the entropy surface. Growth of large solitary wave (${\omega }_s<\mu <0$) increases the entropy (blue curve), while for small solitary wave ($\mu <{\omega }_s<0$) it is decay that increases the entropy (red curve). The tangential point ${\omega }_s=\mu =-0.1$ corresponds to an unstable equilibrium of the solitary and the thermal waves.

Figure 7

Time evolution of the spatial maxima of ${\left|\varphi (x,t)\right|}^2$ for a solitary wave with the initial frequency ${\omega }_s=-0.1$ and an initial squared amplitude $|{\varphi }_m{|}^2=0.238$ (corresponding to the smaller solitary wave in Fig. 3) immersed in thermal waves with various temperatures and chemical potentials. The temperatures are chosen so the average squared amplitude is $\langle |{\varphi }_w{|}^2\rangle =0.012$ for all simulations. The lines are running time averages over 2000 time units. The ten blue lines correspond to thermal waves with chemical potentials from $\mu =0$ to $\mu =-0.09$, the black line corresponds to $\mu =-0.1$, and the ten red lines correspond to values from $\mu =-0.11$ to $\mu =-0.2$. Increase indicates growth of the solitary waves which approach the shape of the bigger structure in Fig. 3. Decrease indicates the the erosion of the initial solitary waves which eventually decay below the amplitudes of the thermal waves.

Figure 8

Spatial maxima $\text{max}\left(\right|\varphi {|}^2)$ of the squared amplitude at $t=4\times {10}^5$ for the simulations of Fig. 7. With the exception of outliers at $\mu =-0.11$ and $\mu =-0.08$, the solitary wave that has an initial frequency ${\omega }_s=-0.1$ grows and approaches the maximum amplitude ${\left|\varphi \right|}^2=3/4$ for $-0.1<\mu \le 0$, and decays and vanishes in the background waves for $\mu <-0.1$.