The deterministic generation of robust soliton comb has significant meaning for the optical frequency combs to be widely used in various applications. As a novel form of microcomb, Soliton crystal holds the advantages of easy generation, high conversion efficiency, and excellent thermal robustness. Here, we report the turnkey deterministic generation of ‘Palm-like’ soliton crystal with a free-running scheme. The robustness of the turnkey soliton crystal generation is also investigated in multiple aspects, including the success rate, the thermal robustness, and the long-term stability. The experiment results reveal our turnkey soliton crystal can achieve nearly a 100% success rate with a power variation less than ±1.5 dB over one hundred trials of two samples, is insensitive to thermal effects, and is robust to the environment during four-hour laboratory time.

This paper secures the analytical soliton solutions of the perturbed NLSE gives to (1). The existence criteria have been demonstrated and the wave speed of the soliton befall out. To achieve all the results obtained we used the mathematical technique. In view of the results obtained, some new additional one have been reported in left-handed metamaterials.

This paper presents the perturbation theory for the double–sine–Gordon equation. We obtain a system of differential equations that shows the amplitude and phase modulation of the approximate solution.. In the particular case λ = 0 we get the well-known perturbation theory for the sine–Gordon equation. For a special value λ=-1/8, we derive a phase-locked solution with the same frequency of the linear case. In general we obtain both coherent (solitary waves, lumps and so on) solutions as well as fractal solutions. We can demonstrate the existence of envelope wobbling solitary waves, because of the phase modulation depending on the solution amplitude and on the position. The main conclusion is that it is too reductive focus only on coherent solutions for the double sine-Gordon equation, because of the very rich behavior for the DSG nonlinear equation, including wobbling chaotic and fractal solutions.

We introduce a model of a passive optical cavity based on the two-dimensional Lugiato-Lefever equation, with a localized pump carrying intrinsic vorticity S, and the cubic or cubic-quintic nonlinearity. Up to S = 5, stable vortex-ring states (vortex pixels) are produced by a variational approximation and in a numerical form. Surprisingly, vast stability areas of the vortex states are found, for both the self-focusing and defocusing signs of the nonlinearity, in the plane of the pump-strength and loss parameters. When the vortex-rings are unstable, they are destroyed by azimuthal perturbations which break the axial symmetry. The results suggest new possibilities for mode manipulations in nonlinear optical media.

This paper aims to search for lump waves in a spatial symmetric (2+1)-dimensional dispersive wave model. Through an ansatz on positive quadratic functions, we conduct symbolic computations with Maple to generate lump waves for the proposed nonlinear model. A line of critical points of the lump waves is computed, whose two spatial coordinates travel at constant speeds. The corresponding maximum and minimum values are evaluated in terms of the wave numbers, and interestingly, all those extreme values do not change with time, either. The last section is the conclusion.

The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: 1) Lax representation $[L,M]=0$; 2)~construction of fundamental analytic solutions (FAS); 3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) $\xi^+(x,t,\lambda) =\xi^-(x,t,\lambda) G(x,t\lambda)$ on a contour $\Gamma$ with sewing function $G(x,t,\lambda)$; 4) soliton solutions and possible applications.Step 1) involves several assumptions: the choice of the Lie algebra $\mathfrak{g}$ underlying $L$, as well as its dependence on the spectral parameter, typically linear or quadratic in $\lambda$. In the present paper we propose another approach which substantially extends the classes of integrable NLEE.Its first advantage is that one can effectively use any polynomial dependence in both $L$ and $M$. We use the following steps: A) Start with canonically normalized RHP with predefined contour $\Gamma$; B) Specify the $x$ and $t$ dependence of the sewing function defined on $\Gamma$; C) Introduce convenient parametrization for the solutions $ \xi^\pm(x,t,\lambda)$ of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); D) use Zakharov-Shabat dressing method to derive their soliton solutions. This needs correctly taking into account the symmetries of the RHP. E) Define the resolvent of the Lax operator and use it analyze its spectral properties.

Discovery of stable and broad frequency combs in monochromatically pumped high-Q optical Kerr microresonators caused by generation of temporal solitons can be regarded as one of the major breakthroughs in nonlinear optics during the last two decades. Transfer of the soliton-comb concept to χ(2) microresonators promises lowering of the pump power, new operation regimes, and entering new spectral ranges; scientifically, it is of a big challenge. Here we represent an overview of stable and accessible soliton-comb regimes in monochromatically pumped χ(2) microresonators discovered during the last several years. The main stress is made on lithium niobate based resonators. The overview pretends to be rather simple, complete, and comprehensive: It incorporates the main factors affecting the soliton-comb generation, such as the choice of the pumping scheme (pumping to the first or second harmonic), the choice of phase matching scheme (natural or artificial), the effects of temporal walk-off and dispersion coefficients, and also the influence of frequency detunings and Q-factors. Most of discovered nonlinear regimes are self-starting – they can be accessed from noise upon a not very abrupt increase of the pump power. The soliton-comb generation scenarios are not universal – they can be realized only under proper combinations of the above mentioned factors. We indicate what kind of restrictions on the experimental conditions have to be imposed to get the soliton-comb generation.

This article is a followup to an earlier review which outlined some of the interesting features of the soliton/wave-action potential (AP) model, and noted the need to test its key aspects; including the need to test if its presumed lipid phase transition is actually happening during AP firings in excitable cells. The intent here is to point out the sort of tests, and evidence from them, that might be needed if the soliton/wave-AP model is to be accepted broadly by biologists. Here, after an overview of the modern electrophysiological-AP model and of the soliton/wave-AP model, there are three areas considered. First, possible compositional influences on membrane properties relative to the soliton/wave-AP model are presented. Including questions with regard to the soliton/wave-AP model’s assumption that changes in surface potentials influence the transmembrane potential. Second, some recent work from the good folks who advocate for the soliton/wave-AP model concerning the occurrence of lipid phase transitions in neurons or in extracts from nervous tissues are examined. Here it is noted that there is a need to consider whether these lipid phase transitions happen within normal physiological conditions or not. Third, and finally, the advocates for the soliton/wave-AP model have adopted a thermodynamic/theory-based philosophical approach in their studies. It is argued that this philosophical approach is a radical departure from the philosophical approach used under the scientific method. The features of this new approach, and implications its use, are examined.

It is introduced and studied para-Ricci-like solitons with potential Reeb vector field on almost paracontact almost paracomplex Riemannian manifolds. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field have been considered. It is proved a necessary and sufficient condition the manifold to admit a para-Ricci-like soliton which is the structure to be para-Einstein-like. Explicit examples are provided in support of the proven statements.

A generalization of $\eta$-Ricci solitons is considered involving an additional metric and functions as soliton coefficients. The soliton potential is torse-forming and orthogonal to the contact distribution of the almost contact B-metric manifold. Then such a manifold can also be considered as an almost Einstein-like manifold, a generalization of an $\eta$-Einstein manifold with respect to both B-metrics and functions as coefficients. Necessary and sufficient conditions are found for a number of properties of the curvature tensor and its Ricci tensor of the studied manifolds. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

The scalar field of space-time film is considered as unified fundamental field. The field model under consideration is the space-time generalization of the model for a two-dimensional thin film. The force and metrical interactions between solitons are considered. These interactions correspond to the electromagnetic and gravitational interactions respectively. The metrical interaction and its correspondence to the gravitational one are considered in detail. The practical applications of this approach are discussed.

Owing to the nondiffracting, self-accelerating, and self-healing properties, Airy beams of different nature have become a subject of immense interest in the past decade. Their interesting properties have opened doors to many diverse applications. Consequently, the questions of how to properly design spatial manipulation of Airy beams or how to implement them in different setups have become important and timely in the development of various optical devices. Here, based on our previous work, we present a short review on the spatial control of Airy beams, including the interactions of Airy beams in nonlinear media, beam propagation in harmonic potential, and the dynamics of abruptly autofocusing Airy beams in the presence of a dynamic linear potential. We demonstrate that under the guidance of nonlinearity and external potential, the trajectory, acceleration, structure, and even the basic properties of Airy beams can be adjusted to suit specific needs. We describe other fascinating phenomena observed with Airy beams, such as self-Fourier transformation, periodic inversion of Airy beams, and the appearance of spatial solitons in the presence of nonlinearity. These results have promoted the development of Airy beams, and have been utilized in various applications, including particle manipulation, self-trapping, and electronic matter waves.

By using Modified simple equation method, we study the Cahn Allen equation which arises in many scientific applications such as mathematical biology, quantum mechanics and plasma physics. As a result, the existence of solitary wave solutions of the Cahn Allen equation is obtained. Exact explicit solutions interms of hyperbolic solutions of the associated Cahn Allen equation are characterized with some free parameters. Finally, the variety of structure and graphical representation make the dynamics of the equations visible and provides the mathematical foundation in mathematical biology, quantum mechanics and plasma physics.

In the present study, we derive a new B-spline technique namely trigonometric B-spline collocation algorithm to solve some initial boundary value problems for the nonlinear Klein-Gordon equation. In order to carry out the time integration with Crank-Nicolson implicit method, the order of the equation is reduced to give a coupled system of nonlinear partial differential equations. The collocation approximation based on trigonometric cubic B-splines for spatial discretization is followed by the linearization of the nonlinear term. The efficiency and accuracy of the present method are validated by measuring the error between the numerical and analytical solutions when exist. The conservation laws representing momentum and energy are also computed for all problems.

Exact and soliton type solutions have great importance in propagation of surface waves, fluid dynamics, optics, and many other elds of nonlinear sciences. In this study, the explicit and exact soliton type solutions for two space-time fractional Equal- Width (FEW) equations with conformable derivative are procured via the hyperbolic function approach. The wave type solutions are represented in some hyperbolic and trigonometric functions.

In this paper, the functional variable method is used to establish solitary wave solutions and periodic wave solutions of the loaded quadratic non-linear Klein-Gordon equation, the loaded cubic non-linear Klein-Gordon equation and the loaded coupled non-linear Klein-Gordon equation. All solutions of these equations have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab software. The main advantage of the proposed functional variable method over other methods is that it provides more new exact traveling wave solutions along with additional free parameters. The graphical representations of the soliton solutions and the periodic wave solutions by using distinct values of random parameter are demonstrated to better understand their physical features. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena.

The basis for computation in the brain is the quantum threshold of the ‘soliton’ accompanying the ion changes of the action potential and the refractory membrane at convergences. We provide a logical explanation from the action potential to a neuronal model of the coding and computation of the retina and have explained how the visual cortex operates by quantum phase processing. In the small world network parallel frequencies collide into definable patterns of distinct objects. Elsewhere we have shown how many sensory cells are mean sampled to a single neuron and that convergences of neurons are common. We have also demonstrated, using the threshold and refractory period of a quantum phase pulse, that action potentials diffract across a neural network due to the annulment of parallel collisions in phase ternary computation (PTC). Thus, PTC applied to neuron convergences results in collective mean sampled frequency and is the only mathematical solution within the constraints of brain neural networks (BNN). In the retina and other sensory areas, we discuss how this information is coded and then understood in terms of network abstracts within the lateral geniculate nucleus (LGN) and visual cortex. First by defined neural patterning within a neural network, and then in terms of contextual networks, we demonstrate that the output of frequencies from the visual cortex contain information amounting to abstract representations of objects in increasing detail. We show that nerve tracts from the LGN provide time synchronisation to the neocortex (defined as the location of the combination of connections of the visual cortex, motor cortex, auditory cortex, etc). The full image is therefore combined in the neocortex with other sensory modalities so that it receives information about the object from the eye, and all abstracts that make up the object. Spatial patterns in the visual cortex are formed from individual patterns illuminating the retina and memory is encoded by reverberatory loops of computational actions potentials (CAPs). We demonstrate that a similar process of PTC may take place in the cochlea and associated ganglia, as well as ascending information from the spinal cord, and that this function should be considered universal where convergences of neurons occur.

Nonlinear compression of narrow-band (Δ ≈ 0.2 cm-1) 20 ns KrF laser pulses in SF6 at 10 atm and CH4 at 50 atm pressure was studied. Both SBS and SRS optically phase conjugated backward reflected radiation was registered with energy reflectivity 10−14% in SF6 and CH4. In SF6 SBS pulses gradually shortened from 10 ns to 2−3 ns with pumping decrease to the SBS threshold ~ 10 mJ, while the SRS pulse had the shortest length of 30−60 ps for the maximal pumping of 120 mJ and broadened near the SRS threshold of ~30 mJ. For the SRS pulse energy ~ 2 mJ peak power 5×107 W was tenfold higher than the pump power. The theoretical model predicted a soliton-like SRS pulse compression to a temporal length of the order of vibrational relaxation time. There was no pulse compression of backward SBS and SRS radiation in CH4, while in the forward direction SRS pulses shortened to 3−4 ns at reduced pumping.

Almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds, are in principle equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized here as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e. its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e. it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi-Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed and the properties obtained in the theoretical part are confirmed.

The nucleation and dynamics of Josephson and Abrikosov vortices determine the critical currents of layered high-T_c superconducting (HTS) thin films, grain boundaries, and coated conductors, so understanding their mechanisms is of crucial importance. Here we treat pair creation of Josephson and Abrikosov vortices in layered superconductors as a secondary Josephson effect, in which each full vortex is viewed as a composite fluid of micro-vortices, such as pancake vortices, which tunnel coherently via a tunneling matrix element. We introduce a two-terminal magnetic (Weber) blockade effect that blocks tunneling below a threshold current, and simulate time-correlated vortex tunneling above threshold. The model shows nearly precise agreement with voltage-current (V-I) characteristics of HTS cuprate grain boundary junctions, which becomes more concave rounded as temperature decreases, and also explains the piecewise linear V-I behavior observed in iron-pnictide bicrystal junctions and other HTS devices. When applied to either Abrikosov or Josephson pair creation, the model explains a plateau seen in plots of critical current vs. thickness of HTS coated conductors. The observed correlation between theory and experiment strongly supports the proposed quantum picture of vortex nucleation and dynamics in layered superconductors.