One-dimensional Nonlinear Lattices Research Articles

Heat transport in an angular-momentum-conserving lattice.

It is expected that the energy-diffusion propagator in a one-dimensional nonlinear lattice with three conserved quantities: energy, momentum, and stretch, consists of a central heat mode and two sound modes. The heat mode follows a Lévy distribution. Consequently, the heat diffusion is super, i.e., the second moment of the diffusion propagator diverges as t^{β} with β>1; and the heat conduction is anomalous, i.e., the heat conductivity is size dependent and diverges with size N by N^{α}, with α>0. In this paper, we study a one-dimensional lattice with two-dimensional transverse motions, in which the total angular momentum also conserves. More importantly, the diffusion of this conserved quantity is ballistic. Surprisingly, the above pictures and the values of the mentioned power exponents keep unchanged. The universality of the scalings is then further extended. On the other hand, the detailed strengths of heat transports are largely enhanced. Such a counterintuitive finding can be explained by the change of the phonon mean-free path of the lattices.

Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution’s quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated.

In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride

This paper analyzes the analytical and numerical solutions’ structure of the combined mKdV equation and KdV equation (mKdV+KdV equation) using the Khater II (Khat. II) method and three accurate B-spline numerical schemes. ExCBS, SBS and TQBS numerical schemes are the numerical systems used. The handled model describes many distinct phenomena such as wave propagation of bounded particles with a harmonic force in a one-dimensional nonlinear lattice, propagation of ion-acoustic waves of small amplitude without Landau damping in plasma physics, and propagation of thermal pulse through a single sodium fluoride crystal in solid physics. Numerous examples show the relationship between quick and slow soliton, which generates phase shift. This phase shift is shown in a contour map to show the modest and colossal energy density along the path of fast and slow colliding solitons. Calculating the difference between analytical and numerical solutions shows whether they match spline-connected and distribution graphs.

New impressive representations for the soliton behaviors arising from the (2+1)-Boussinesq equation

In the present work, we will detect unpredicted conducts “which weren't realized before” to the soliton solutions of the (2+1)-Boussinesq equation by using three distinct algorithms. The proposed equation is the famous nonlinear one which distinguishes the waves of coastal and ocean engineering that involve the nonlinearity and dispersion terms. Moreover, it is a developed form of the standard Boussinesq equation which describes the solution interaction mechanism of shallow-water waves that involve many waves and shallow water influences refraction, diffraction, shoaling and weak nonlinearity properties arising in fluid dynamics. It also plays principal role in many physics branches, such as propagation of long waves in shallow water, vibrations in a nonlinear string and ion sound waves in plasma and one-dimensional nonlinear lattice waves. The three algorithms that will select for this purpose are the (G′G) -expansion method, the extended direct algebraic method (EDAM) and the extended simple equation method (ESEM). These three distinct manners are applied in the same time and parallel. We will show comparison between our new soliton solution behaviours with that constructed before.

An efficient computational technique for class of generalized Boussinesq shallow-water wave equations

In this study, a modification of the homotopy perturbation method called the homotopy perturbation transform method (shortly, HPTM) is applied to the generalized Boussinesq and generalized perturbed Boussinesq equations. These equations depict the propagation of long waves in shallow water under gravity, one-dimensional nonlinear lattices, and nonlinear strings. They are also significantly important in coastal and ocean engineering. The numerical simulations have been implemented to verify that the proposed scheme is very reliable and accurate. Significant results and their graphical depictions revealed that HPTM is more appropriate for nonlinear problems, and it is immensely active for nonlinear models of classical and fractional order type. The obtained results illuminate that the proposed method is simple to implement and very effective to analyze the complex models arising in science, technology, and engineering.

Rogue and solitary waves in coupled phononic crystals.

In this work we present an analytical and numerical study of rogue and solitary waves in a coupled one-dimensional nonlinear lattice that involves both axial and rotational degrees of freedom. Using a multiple-scale analysis, we derive a system of coupled nonlinear Schrödinger-type equationsin order to approximate solitary waves and rogue waves of the coupled lattice model. Numerical simulations are found to agree with the analytical approximations. We also consider generic initialization data in the form of a Gaussian profile and observe that they can result in the spontaneous formation of rogue-wave-like patterns in the lattice. The solitary and rogue waves in the lattice demonstrate both energy isolation and exchange between the axial and rotational degrees of freedom of the system. This suggests that the studied coupled lattice has the potential to be an efficient energy isolation, transfer, and focusing medium.

Heat transport in nonlinear lattices free from the umklapp process.

We construct one-dimensional nonlinear lattices having the special property such that the umklapp process vanishes and only the normal processes are included in the potential functions. These lattices have long-range quartic nonlinear and nearest-neighbor harmonic interactions with/without harmonic onsite potential. We study heat transport in two cases of the lattices with and without harmonic onsite potential by nonequilibrium molecular dynamics simulation. It is shown that the ballistic heat transport occurs in both cases, i.e., the scaling law κ∝N holds between the thermal conductivity κ and the lattice size N. This result directly validates Peierls's hypothesis that only the umklapp processes can cause the thermal resistance while the normal ones do not.

The classification of the single traveling wave solutions to the KdV equation with higher-order nonlinearity

The Korteweg–de Vries (KdV) equation with higher-order nonlinearity is considered as the model of the wave propagation in one-dimensional nonlinear lattice. In this paper, we obtain the exact solutions to the KdV equation with higher-order nonlinearity. The complete discrimination system for polynomial method is applied to handle such a problem. All the possible exact traveling wave solutions are obtained.

Nonlinear edge modes from topological one-dimensional lattices

We propose a method to address the existence of topological edge modes in one-dimensional (1D) nonlinearlattices, by deforming the edge modes of linearized models into solutions of the fully nonlinear system. Forlarge enough nonlinearites, the energy of the modified edge modes may eventually shift out of the gap, leadingto their disappearance. We identify a class of nonlinearities satisfying a generalised chiral symmetry wherethis mechanism is forbidden, and the nonlinear edge states are protected by a topological order parameter.Different behaviours of the edge modes are then found and explained by the interplay between the nature of thenonlinarities and the topology of the linearized models.

Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

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 Schrödinger, where the linear operator of the Schrödinger 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 presence of a cubic nonlinearity leads to the emergence of embedded-in-the-continuum discrete breathers. In both settings, it is seen that the stability of the modes near the linear limit turns into instability as nonlinearity is increased past a critical value, leading to a dynamical delocalization of the solitary wave (or breathing) state. Finally, we suggest a concrete experiment to observe these embedded modes using a bi-inductive electrical lattice.

Fractional nonlinear electrical lattice.

We examine the linear and nonlinear modes of a one-dimensional nonlinear electrical lattice, where the usual discrete Laplacian is replaced by a fractional discrete Laplacian. This induces a long-range intersite coupling that, at long distances, decreases as a power law. In the linear regime, we compute both the spectrum of plane waves and the mean-square displacement (MSD) of an initially localized excitation, in closed form in terms of regularized hypergeometric functions and the fractional exponent. The MSD shows ballistic behavior at long times, MSD∼t^{2} for all fractional exponents. When the fractional exponent is decreased from its standard integer value, the bandwidth decreases and the density of states shows a tendency towards degeneracy. In the limit of a vanishing exponent, the system becomes completely degenerate. For the nonlinear regime, we compute numerically the low-lying nonlinear modes, as a function of the fractional exponent. A modulational stability computation shows that, as the fractional exponent decreases, the number of electrical discrete solitons generated also decreases, eventually collapsing into a single soliton.

Approach to Phonon Relaxation Time and Mean Free Path in Nonlinear LatticesSupported by the National Natural Science Foundation of China (Grant Nos. 12075199 and 11675133).

Based on the self-consistent phonon theory, the spectral energy density is calculated by the canonical transformation and the Fourier transformation. Through fitting the spectral energy density by the Lorentzian profile, the phonon frequency as well as the phonon relaxation time is obtained in one-dimensional nonlinear lattices, which is validated in the Fermi–Pasta–Ulam-β (FPU-β) and ϕ 4 lattices at different temperatures. The phonon mean free path is then evaluated in terms of the phonon relaxation time and phonon group velocity. The results show that, in the FPU-β lattice, the phonon mean free path as well as the phonon relaxation time displays divergent power-law behavior. The divergent exponent coincides well with that derived from the Peierls–Boltzmann theory at weak anharmonic nonlinearity. The value of the divergent exponent expects a power-law divergent heat conductivity with system size, which violates Fourier’s law. For the ϕ 4 lattice, both the phonon relaxation time and mean free path are finite, which ensures normal heat conduction.

Energy and momentum diffusion in one-dimensional periodic and asymmetric nonlinear lattices with momentum conservation.

The energy diffusion in one-dimensional (1D) momentum conserving nonlinear lattices usually exhibits anomalous superdiffusion, except the coupled rotator lattice with symmetric and periodic interacting potential which has normal energy diffusion corresponding to normal heat conduction. For nonperiodic 1D lattices with momentum conservation, it has been argued that the asymmetric potential can induce normal heat conduction. Later results indicate the observed normal behavior might be the finite size effect and the anomalous behavior will appear in the thermodynamical limit. Here we propose asymmetric and periodic 1D nonlinear lattices with momentum conservation. The energy and momentum diffusion behaviors will be investigated in detail and the same normal diffusion behaviors for both energy and momentum can be observed. These results confirm that the periodicity is the key for normal transport behavior in 1D momentum conserving lattices, whether the potential is symmetric or asymmetric.

Induced localized nonlinear modes in an electrical lattice

Intrinsic localized modes, also called discrete breathers (DBs), can exist under certain conditions in one-dimensional nonlinear electrical lattices driven by external periodic excitations (PEs). In this work, we have studied experimentally the effectiveness of some PEs of variable waveform at generating DBs in such lattices. We have found that this generation phenomenon is optimally controlled by the strength of the fundamental harmonic of the driving.

Scaling property of the heat-current flows across a weak interface

We systematically study heat current $J$ that flows through a few one-dimensional nonlinear lattices, each of which consists of two identical segments that are coupled by a weak interface. Existing theoretical analyses expect that $J$ is generally proportional to the square of the interface strength when the temperature drop is fixed and small. However, we observe two completely different classes in our numerical simulations. One follows the original expectation. In the other class, however, $J$ follows a power-law decay with the strength and the detailed power exponent depends on the details of the lattices and the interface interaction. Further theoretical analyses reveal that in the former class the interface potential energy decays with the interface strength linearly, which is commonly observed. In the latter class, the interface potential energy approaches a constant that is independent of the interaction strength. The detailed power exponents are then well explained.

Lie point symmetries, conservation laws, and analytical solutions of a generalized time-fractional Sawada–Kotera equation

ABSTRACTUnder investigation in this work is the invariance properties of the time-fractional generalized Sawada–Kotera equation, which can describe motion of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice. With the help of the Lie symmetry analysis method of fractional differential equations, we strictly derive the vector fields and symmetry reductions of the equation. Furthermore, based on the power series theory, an effective method is presented to succinctly construct its analytical solutions. Finally, using the new conservation theorem, the conservation laws of the equation are well constructed with a detailed derivation.

Renormalized vibrations and normal energy transport in 1d FPU-like discrete nonlinear Schr\xf6dinger equations

For one-dimensional (1d) nonlinear atomic lattices, the models with on-site nonlinearities such as the Frenkel-Kontorova (FK) and ϕ4 lattices have normal energy transport while the models with inter-site nonlinearities such as the Fermi-Pasta-Ulam-β (FPU-β) lattice exhibit anomalous energy transport. The 1d Discrete Nonlinear Schrödinger (DNLS) equations with on-site nonlinearities has been previously studied and normal energy transport has also been found. Here, we investigate the energy transport of 1d FPU-like DNLS equations with inter-site nonlinearities. Extended from the FPU-β lattice, the renormalized vibration theory is developed for the FPU-like DNLS models and the predicted renormalized vibrations are verified by direct numerical simulations same as the FPU-β lattice. However, the energy diffusion processes are explored and normal energy transport is observed for the 1d FPU-like DNLS models, which is different from their atomic lattice counterpart of FPU-β lattice. The reason might be that, unlike nonlinear atomic lattices where models with on-site nonlinearities have one less conserved quantities than the models with inter-site nonlinearities, the DNLS models with on-site or inter-site nonlinearities have the same number of conserved quantities as the result of gauge transformation.

Response and correlation functions of nonlinear systems in equilibrium states.

In this paper, we study systematically a serial of correlation functions in some one-dimensional nonlinear lattices. Due to the energy conservation law, they are implicitly interdependent. Various transport coefficients are thus also connected. In the studies of the autocorrelations of local energy density and of local heat current, a general relation between diverging heat conduction and super heat diffusion has been proposed recently. We clarify that such a relation is valid only in systems without temperature pressure. In those with temperature pressure, a constant but nontrivial term appears. This term explains a previously observed fact that heat diffusion in such systems is always ballistic but heat conduction can diverge very slowly. Such a result not only disproves the existence of any general relation between diverging heat conduction and super heat diffusion, but it also breaks the long-term presumption that ballistic heat conduction and diffusion always coexist.

Resonance phonon approach to phonon relaxation time and mean free path in one-dimensional nonlinear lattices.

We extend a previously proposed resonance phonon approach that is based on the linear response theory. By studying the complex response function in depth, we work out the phonon relaxation time besides the oscillating frequency of the phonons in a few one-dimensional nonlinear lattices. The results in the large wave-number-k regime agree with the expectations of the effective phonon theory. However, in the small-k limit they follow different scaling laws. The phonon mean free path can also be calculated indirectly. It coincides well with that derived from the anharmonic phonon approach. A power-law divergent heat conduction, i.e., the heat conductivity κ depends on lattice length N by κ∼N^{β} with β>0, then is supported for the momentum-conserving lattices. Furthermore, this approach can be applied to diatomic lattices. So obtained relaxation time quantitatively agrees with that from the effective phonon theory. As for the mean free path, the resonance phonon approach can detect both the acoustic and the optical branches, whereas the anharmonic phonon approach can only detect a combined branch, i.e., the acoustic branch for small k and the optical branch for large k.

Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation

The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalizedexp⁡(-Φ(ξ))-expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations.