11.1 Coherent sampling of dissipative Kerr **soliton** dynamics. (a) Concep- tual schematic showing microresonator signal (red) combined with the probe sampling pulse train (blue) using a bidirectional coupler. The probe pulse train repetition rate is offset slightly from the mi- croresonator signal. It temporally samples the signal upon photo detection to produce an interferogram signal shown in the lower panel. The measured interferogram shows several frame periods during which two solitons appear with one of the solitons experienc- ing decay. (b) Left panel is the optical spectrum and right panel is the FROG trace of the probe EO comb (pulse repetition period is shown as 46 ps). An intensity autocorrelation in the inset shows a full-width-half-maximum pulse width of 800 fs. (c) Microresonator pump power transmission when the pump laser frequency scans from higher to lower frequency. Multiple ‘steps’ indicate the formation of solitons. (d) Imaging of **soliton** formation corresponding to the scan in panel (c). The x-axis is time and the y-axis is time in a frame that rotates with the solitons (full scale is one round-trip time). The right vertical axis is scaled in radians around the microresonator. Four **soliton** trajectories are labeled and fold-back into the cavity coordi- nate system. The color bar gives their signal intensity. (e) **Soliton** intensity patterns measured at four moments in time are projected onto the microresonator coordinate frame. The patterns correspond to initial parametric oscillation in the modulation instability (MI) regime, non-periodic behavior (MI regime), four **soliton** and single **soliton** states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11.2 Measurements of non-repetitive **soliton** events. (a) Two solitons

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is a well-known mathematical model for describing the evolution of pulses in **nonlinear** optical ﬁbers and of surface gravity waves in ﬂuid dynamics []. To investigate diﬀerent complex **nonlinear** phenomena of our realistic world, some generalizations of the widely used ( + )-dimensional Eq. () into a ( + )-dimensional one are obtained [–]. Theo- retical and experimental research of these higher-dimensional integrable generalizations has been carried out due to their attraction and application in many ﬁelds such as plasma **physics**, **nonlinear** optics, ﬂuid dynamics, and Bose-Einstein condensates [–]. The NLS equation admits the following ( + )-dimensional extension:

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Systems of NLS equations arise in many fields of **physics**, including condensed matter, hydrodynamics, optics, plasmas, and Bose-Einstein condensates (BECs) (see e.g. [1], [4], [12], [23]). In the presence of strong periodic trapped potentials, a NLS equation can be approximated by a DNLS equation by using the “tight binding approximation” [24]. Equation (1) describes a large class of discrete **nonlinear** systems such as optical fibers [5], [6], small molecules such as benzene [7], and, more recently, dilute BECs trapped in a multiwell potential [2], [3], [25], [24].

Parallel developments in both computer technologies and symbolic softwares have greatly contributed to solve lots of problems defined in various fields covering applied mathematics, **physics** and many engineering fields. A diverse class of effective methods have successfully been introduced to study this class of equations, for example [3, 10–15]. On the other hand some of the commonly used approaches, for solving **nonlinear** evolution equations, are: The ansatz [16–18], modified simple equation [19], the first integral [20,21], ( G G 0 )-expansion [22], sine-Gordon expansion [23, 24]. Furthermore, some other excellent works like Kudryashov methods [25], a modified form of Kudryashov and functional variable methods [26–28] have been done by different researchers. In [29–32], the auxiliary equation, the improved tan( φ(η) 2 )-expansion methods and the exp function approach have been explored for discrete and fractional order PDEs as well. Ali and Hassan [33], Hosseini et al. [34] and Zayed and Al-Nowehy [35] all have utilized the exp a function method to explore the exact solutions

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All these activities, diverse and complementary, made in this field [1-14], are also mainly motivated by the wide roles of Einstein and Dirac equations in modern **physics**, for example, for investigating the spin particle and for the necessity of analysis of synchrotronic radiation [11]. To this purpose, many systems have been subjects of considerable interest and studies. The pioneering inves- tigation could be the work by Drill and Wheeler in 1957 [3], who considered the Dirac equation in a central gravitational field associated with a diagonal metric. Us- ing a normal diagonal tetrad, these authors constructed the generalized angular momentum operator separating the variables in the Dirac equation. Later, in a remarkable paper, appeared in 1987 [12], entitled “Criteria of sepa- rability of variables in the Dirac equation in gravita- tional fields”, Shishkin and Andrushkevich provided the necessary and sufficient conditions, based on rigorous theorems, for separability of the variables for a diagonal tetrad gauge, and deduced the operators that determine the dependence of the wave function on the separated variables. In the same year, Barut and Duru [10] gave exact solutions of the Dirac equation in spatially flat Robertson-Walker space-times for models of expanding universes and discussed the current decomposition. Hence- forth the investigations go into diverse directions, con- sidering various classes of models including different metrics, the general class of which is investigated by Hounkonnou and Mendy in 1999 [13]. Thus, for example, the usual Friedman-Lemaître-Roberston-Walker homoge- neous and isotropic metric of standard cosmology be- longs to this general class of metrics (whether in Carte- sian or spherical coordinates), which also includes gen- eral classes of Kantowski-Sachs metrics for anisotropic cosmologies as well as some examples of metrics used in models for stellar gravitational collapse [14]. It may be worth pointing out that a priori, this class of metrics solves Einstein’s equations for specific distributions of energy-momentum of matter in space-time, in the pres- ence of which the study of the quantized Dirac field may be of interest. Such an avenue could be pursued. For de- tails, see [13] and references therein.

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The **nonlinear** partial differential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in **physics**, chemistry, biology, fluid dynamics, plasma, optical fibers and other areas of engineering. Many efforts have been made to study NPDEs. One of the most exciting advances of **nonlinear** science and theoretical **physics** has been a development of methods that look for exact solutions for **nonlinear** evolution equations. The availability of symbolic computations such as Mathematica or Maple, has popularized direct seeking for exact solutions of **nonlinear** equations. Therefore, exact solution methods of **nonlinear** evolution equations have become more and more important resulting in methods like the tanh method [1–3], extended tanh function method [4, 5], the modified extended tanh

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The **nonlinear** Schrödinger (NLS) equation is one of the important mathematical mod- els in many ﬁelds of **physics**, which has been widely applied in Bose-Einstein condensates [–], **nonlinear** optical ﬁber communication [, ], plasma **physics** [, ], hydrodynamics [], and so on. Recently more and more people have been devoted to solving the exact solutions of the generalized NLS models [–]. Today, the temporal optical solitons of the NLS equation have been the objects of theoretical and experimental studies in opti- cal ﬁber communication, and optical solitons are regarded as an important alternative to the next generation of ultrafast optical telecommunication systems. The study of optical solitons has reached the stage of a real-life application. The propagation of optical pulse in monomode optical ﬁber is governed by the NLS equation.

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The presence of an interface between different materials can profoundly affect the evolution of **nonlinear** excitations. Such interface can support stationary surface waves. These were encountered in various areas of **physics** including solid-state **physics** [1], near surface optics [2], plasmas [3], and acoustics [4]. In **nonlinear** optics, surface waves were under active consideration since 1980. The progress in their experimental observation was severely limited because of unrealistically high power levels required for surface wave excitation at the interfaces of natural materials. However, shallow refractive index modulations accessible in a technologically fabricated waveguide array (or lattice) may facilitate the formation of surface waves at moderate power levels at the edge of semi-infinite arrays as was suggested in Ref. [5]. This has led to the observation of one-dimensional surface solitons in arrays with focusing nonlinearity [6]. Defocusing lattice interfaces are also capable to support surface gap solitons [7, 8 & 9]. Surface lattice solitons may exist not only in cubic and saturable materials, but also in quadratic [10] and nonlocal [11] media, as well as at the interfaces of complex arrays [12].

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The **nonlinear** partial differential equations of mathematical **physics** are major subjects in physical science [1]. Exact solutions for these equations play an important role in many phenomena in **physics** such as fluid mechanics, hydrodynamics, Optics, Plasma **physics** and so on. Recently many new approaches for finding these solutions have been proposed, for example, tanh - seen method [2]-[4], extended tanh - method [5]-[7], sine - cosine method [8]-[10], homogeneous balance method [11, 12],F-expansion method [13]-[15], exp-function method [16, 17], trigonometric function series method [18], ( 𝐺

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It is well established now that the higher-dimensional **nonlinear** wave ﬁelds have richer be- havior than one-dimensional ones. It was veriﬁed that the existence of two solitons having the structures peculiar to a higher-dimensionality may contribute to the variety of the dy- namics of **nonlinear** waves [1–3]. Thereby, seeking for exact solution and studying dynam- ical behavior [4–7] of solutions are very signiﬁcant in **physics**, mathematics, and nonlin- ear science ﬁelds for understanding the complexity and variety of dynamics determined by high-dimensional **nonlinear** evolution equation [8–10]. In **soliton** theory, the **soliton** solutions are obtained by the use of the inverse scattering method, Bäcklund transfor- mation, Darboux transformation, Painlevè method, Hirota method, the tanh method, the generalized Riccati equation expansion method, homoclinic test method, etc. [11–18]. In this work, we would like to use the parameter perturbation method for seeking dynamical feature of **soliton** solution for the (3 + 1)-dimensional Yu–Toda–Sasa–Fukuyama (YTSF) equation.

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In recent years, many powerful methods to construct exact solutions of **nonlinear** partial diﬀerential equations have been established and developed, which led to one of the most exciting advances of **nonlinear** science and theoretical **physics**. Particularly, the existence of **soliton**-type solutions for **nonlinear** models is of great importance because of their po- tential application in many **physics** areas such as **nonlinear** optics, plasmas, ﬂuid mechan- ics, condensed matter and many more. Remarkably, the interest in dark and bright solitons has been growing steadily in recent years [–]. In fact, many kinds of exact **soliton** solu- tions have been obtained by using, for example, the tanh-sech method [–], extended tanh method [–], homogeneous balance method [, ], ﬁrst integral method [, ], Jacobi elliptic function method [, ], ( G

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[2] N. A. Kudryashov, “Exact **Soliton** Solutions of the Gene- ralized Evolution Equation of Wave Dynamics,” Journal of Applied Mathematics and Mechanics, Vol. 52, No. 3, 1988, pp. 361-365. doi:10.1016/0021-8928(88)90090-1 [3] J. Weiss, “The Panleve Property for Partial Differential

The rest of the paper is organized as follows. In Sect. 2, the bilinear form of (3 + 1)- dimensional potential-YTSF equation is given by applying the Hirota bilinear method. In Sect. 3, N-**soliton** solutions are presented by using the perturbation approach. In Sect. 4, by virtue of the Riemann theta function, periodic wave solutions are derived successfully, and the asymptotic properties of periodic wave solutions show that periodic wave solution degenerate to **soliton** solution. Finally, the concluding remarks are presented in Sect. 5. 2 Bilinear form for (3 + 1)-dimensional potential-YTSF equation

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Here, we demonstrate micro-comb laser cavity-solitons. Laser cavity-solitons are intrinsically background free and have underpinned key breakthroughs in semiconductor lasers [22,25-28]. By merging their properties with the **physics** of multi-mode systems [29], we provide a new paradigm for **soliton** generation and control in micro-cavities. We demonstrate 50 nm wide bright **soliton** combs induced at average powers more than one order of magnitude lower than the Lugiato-Lefever **soliton** power threshold [22], measuring a mode efficiency of 75% versus the theoretical limit of 5% for bright Lugiato-Lefever solitons [23]. Finally, we can tune the repetition- rate by well over a megahertz without any active feedback.

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Critical for micro-combs is the ability to phase-lock the frequency comb modes, and many oscillation states have been explored to achieve this, including feedback-stabilized Kerr combs [s1], dark solitons [s2] and dissipative Kerr solitons (DKS) [s3]. While many of these approaches have enabled breakthroughs [27], all (particularly DKS and dark **soliton** states described by the Lugiato-Lefever equation [27]) require sophisticated feedback systems and complex dynamic pumping schemes to initiate and sustain [s1, s3]. Here, we employ a new and powerful class of micro-comb based on what have been termed “**soliton** crystals” that are generated from a fundamentally different process and which offer significantly improved simplicity compared to DKS states. They are naturally formed in micro-cavities that display the appropriate form of mode crossings, without the need for the complex dynamic tuning mechanisms that DKS require. They were termed ‘**soliton** crystals’, due to their crystal-like profile in the angular domain in micro-ring resonators [36, 37]. To generate coherent micro-combs, a CW pump laser (Yenista Tunics – 100S-HP) was employed, with the power amplified to 30dBm by an optical amplifier (Pritel PMFA-37) and the wavelength subsequently swept from blue to red. The acquired **soliton** crystals optical spectra are shown in Fig. S1. We note that when locking the pump wavelength to the resonance of the MRR, the stability of the microcomb can be further enhanced that could even serve as frequency standards [22].

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Abstract This paper uses the exp( −Φ(ξ ))-Expansion method to investigate solitons to the M-fractional **nonlinear** Schr¨odingers equation with cubic nonlinearity. The results obtained are dark solitons, trigonometric function solutions, hyperbolic solutions and rational solu- tions. Thus, the constraint relations between the model coefficients and the traveling wave frequency coefficient for the existence of solitons solutions are also derived.

In this section, we discuss the **nonlinear** interaction between intense electromagnetic radiation and quantum plasma oscillations [46]. We consider a one-dimensional geometry of an unmagnetized dense electron-ion plasma, in which immobile ions form the neutralizing background. Thus, these phenomena are on a timescale shorter than the ion plasma period. An intense circularly polarized electro- magnetic (CPEM) plane wave interacts nonlinearly with the EPOs, giving rise to an envelope of the CPEM vector potential A ⊥ = A ⊥ ( ˆ x + iˆ y ) exp( − i ω 0 t + ik 0 z), which obeys

Advancement in science and mathematical modelling dealing with complex natu- ral phenomenon give rise to induction of more and more **nonlinear** partial differential equations(NLPDE) with constant and variable coefficients, more precisely with con- cept of deterministic chaos, natural world has been revealed as **nonlinear** one [1]. Due to this inherited nonlinearity in partial differential equation, the integrability is big issue, and moreover the integrability for **nonlinear** partial differential equation cannot be claimed in general. When one says the model is integrable, one should point out under what special meaning it is integrable. B¨ acklund transformations, Lax pairs, conservation laws and infinite symmetries may regarded as predictors of complete in- tegrability [23]. For example, absence of movable singularities in solution of equation can be claimed as integrability in sense of Painlev´e property and equation having multi-**soliton** solution is also eligible candidate for complete integrability in term of Hirota’s formalism.

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