Although the origin of **nonlinear** partial differential equations (nPDEs) is very old, they have undergone remarkable new **developments** during the last half of the twenty century. One of the main impulses for developing nPDEs has been the study of **nonlinear** wave propagation problems. These problems arise in different areas of applied mathematics, physics, engineering, including fluid dynamics, **nonlinear** optics, solid mechanics, plasma physics, field theories, and condensed matter theories to mention some practical examples.

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wave is a wave whose height exceeds the significant wave height of measured wave train by factor more than 2.2 [1] and [2]. Occurrences (where and when) of this wave are not easy to predict, but its impact can cause damage to oceanic objects, i.e. ships and marine structures, that are around this wave (see Earle [3], Mori et al [4], Divinsky and Levin [5], Truslen and Dysthe [6], Smith [7], Toffoli and Bitner [8] and Waseda et al. [9]). Therefore, information about the presence of freak waves is important for offshore activities. The presence has been often reported in media. Nikolkina and Didenkulova [10] collected and analysed freak waves reported in media in 2006-2010. To understand the occurrence, propagation and generation of the extreme wave, various studies have been conducted by many researchers. Waseda et al. [11] conducted deep water observation of freak waves in the North West Pacific Ocean. Hu et al. [12] studied numerically rogue wave based **on** **nonlinear** Schrodinger breather solutions under finite water depth. Islas and Schober [13] investigated the effects of dissipation **on** the development of rogue waves and down shifting by adding **nonlinear** and linear damping terms to the one-dimensional Dysthe **equation**. Xu et al. [14] proposed (2 + 1)- dimensional Kadomtsev–Petviashvili **equation**, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solu- tion to **nonlinear** **evolution** **equation** (NEE). The wave ampli- fication in the framework of forced non linear Schrodinger **equation** is observed by Slunyaev et al. [15]. Cahyono et al. [16] discussed multi-parameters perturbation method for dispersive and **nonlinear** partial differential equations. Ramli [17] investigated **nonlinear** **evolution** of wave group with three frequencies using **third** **order** approximation of Ko- rteweg de Vries **equation** and Maximal Temporal Amplitude. Wabnitz et al. [18] observed extreme wave events which are generated in the modulationally stable normal dispersion regime. Peric et al. [19] regarded a prototype for spatio- temporally localizing rogue waves **on** the ocean caused by **nonlinear** focusing and analyzed by direct numerical simulations based **on** two phase Navier–Stokes equations. Blackledge [20] found the explicit freak waves can not be obtained by pure intuition or by elementary calculations because of their complications. Onorato et al. [21] discussed rogue waves occurring in different physical contexts and related anomalous statistics of the wave amplitude, which deviates from the Gaussian behavior that were expected for random waves. Extreme wave generation using self- correcting method was studied by Fernandez et al. [22]. Xi- eng et al. [23] simulated the extreme wave generation are carried out by using the volume of fluid (VOF) method.

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This combined numerical and experimental study of non- linear wave trains also clarifies the limitations of possible agreement between fully **nonlinear** solution and experiment. We note that while the periodicity in the time domain is pos- sible for propagating and evolving unidirectional waves, they are, strictly speaking, aperiodic in space. This point adds an additional aspect to essential differences that exist between the spatial and temporal formulations of the wave evolu- tion problem, as discussed above. We therefore believe that all **nonlinear** solutions based **on** spatially periodic boundary conditions, as in the method adopted here, as well as in a variety of alternative methods that employ spatial discrete Fourier decomposition, contain intrinsic inaccuracy. These numerical solutions thus can only provide approximate re- sults and require careful experiments to verify their valid- ity. The **present** study shows that the fully **nonlinear** solution, although flawed, yields better agreement with experiments than the application of the spatial version of the modified **nonlinear** Schrödinger (Dysthe) **equation** limited to the **third** **order** that does not require spatial periodicity (Shemer and Alperovich (2013).

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In the past, much of the work in the control of hydraulic systems has used linear model [3] or local linearization of the **nonlinear** dynamics about the nominal operating point [4]. Suitable adaptive approaches are employed when there is no knowledge of the parameter values [5], [6]. In **order** to take system uncertainties into account, robust approaches can be adopted [7], [8]. In [9], a sliding mode control

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The Schrödinger **equation** being a partial differential **equation** describes the quantum state of a physical system and the changes in that system with respect to time. It was first time formulated by Schrödinger in 1925(Schrödinger, 1926) . In the sense of classical mechanics, the governing **equation** predicts the behavior of a system mathematically at any time after the initial state of the system is set and this corresponds to the Newton’s law F ma . In the language of quantum mechanics, the Schrödinger **equation** is analogous to Newton’s law for quantum mechanical system (which usually involves molecules, atoms, sub-atomic particles, whether moving freely, bounded or localized). It is not as simple algebraic operation but in general form it is a linear partial differential **equation** that describes the evaluation of time of the wave function system (Griffiths, 2004).

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(1910) that all equations of this type whose solutions do not have movable critical points (but are allowed to have fixed singular points and movable pole) can be reduced to 50 classes of equations. Moreover, 44 classes out of them are in- tegrable by quadrature or admit reduction of **order**. The remaining 6 equations are irreducible; these are known as the Painlev´e equations or Painlev´e transcendent, and their solutions are known as the Painlev´e transcendental functions. It is significant that the Painlev´e equations often arise in mathematical physics. Some connections are given as follows:

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From the historical **evolution** of the **Third**-front construction policy we can see that the **evolution** of Chinese Communist Party’s regional development theory influences the **third** construction policy and historical process (Zhong, 2011) (仲海涛, 2011). Zhong Haitao (2011) points out that the social, historical, political and economic conditions at home and abroad varies at different periods. So do the ideas **on** China's regional development. The **Third**-front construction largely represents Chinese Communist Party’s ideas **on** regional development.

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A novelty of the **present** study is to **present** an analytical method for investigate dynamic re- sponse of imperfect FGM circular cylindrical shells reinforced by FGM stiffener system and filled inside by an elastic foundations, in thermal environments. Theoretical formulations in terms of dis- placement components according to Reddy’s **third**-**order** shear deformation shell theory (2004) and the smeared stiffeners technique are derived. The thermal elements of shells and stiffeners are taken into account in two cases which are uniform temperature rise law and **nonlinear** temperature change. The closed-form expressions for determining the natural frequency, **nonlinear** frequency- amplitude curve and **nonlinear** dynamic response are obtained by using Galerkin method and fourth-**order** Runge-Kutta method. The effects of stiffener, temperature, foundation, material and dimensional parameters, pre-existent axial compressive and **on** the stability of stiffened FGM shells are considered.

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4. Andres, J., 1986. Boundedness results for solutions of the **equation** &&& x + ax && + g(x)x & + h(x) = p(t) without the hypothesis h(x)sgnx ≥ 0 for x > R . Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 80: 533-539. DOI: MR0976947(89m:34043). 5. Bereketoglu, H. and Györi, I.; 1997. **On** the

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The theory of **nonlinear** difference equations (including discrete Hamiltonian systems) has been widely used to study discrete models in many fields such as computer science, economics, neural networks, ecology and so **on**. Many scholars studied the qualitative properties of difference equations such as stability, oscillation and boundary value problems (see e.g. [1, 2, 3] and references cited therein). But results **on** periodic solutions of difference equations are relatively rare and the results usually obtained by analytic techniques or various fixed point theorems (see e.g. [4]). We may think of (1.1) as being a discrete analogue of the following fourth-**order** Hamiltonian system

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In this paper we derive second and **third** **order** **nonlinear** difference equations for one of the recurrence coefficients in the three term recurrence relation of polyno- mials orthogonal with respect to a modified Laguerre weight. We show how these equations can be obtained from the B¨ acklund transformations of the **third** Painlev´ e **equation**. We also show how to use **nonlinear** difference equations to derive a few terms in the formal asymptotic expansions in n of the recurrence coefficients.

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In this note, periodic analytical approximations for the exact solution of the pendulum **equation** of motion are proposed. As a first approximate solution, I have modi- fied the harmonic approximation proposed in Ref. [17], which resulted in an accurate approximation for ampli- tudes up to π/3 rad. However, due to the unavoidable loss of harmonicity with the increase of amplitude, the harmonic approximation soon becomes poor. In fact, for amplitudes above π/2 rad, which are of interest for some experiments [4, 8, 26–28], the deviations are significant, as shown in Fig. 3. Then, I have found it natural to take the periodicity of the exact solution into account to derive an analytical Fourier series expansion, giving continuity to a numerical treatment I have developed (with co-authors) in a very recent work [24]. This task re- vealed many complexities due to the inverse sine function **present** in the exact solution, our Eq. (16), so I decided to expand only the (periodic) Jacobi elliptic function sn(u; k), rather than the whole function θ(t). The trunca- tion of this series to only a few terms has revealed itself as a good approximation technique, yielding relative er- rors smaller than 0.4% even for amplitudes as large as

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Abstract - The main objective of this study was to evaluate the effect of the drying process **on** the vitamin C levels and physical properties of dedo-de-moça pepper. The drying kinetics and the structural properties were determined as a function of moisture content. Convective drying was compared with freeze-drying in terms of product quality, structural properties, retention of vitamin C and rehydration characteristics. Empirical and semi-empirical equations were used to describe the drying and rehydration kinetics. **Nonlinear** analysis applied to results of convective drying, based **on** curvature measures and bias measures, showed that the only **equation** that gives good inference results based **on** least squares estimators is the Overhults **equation**. The characterization of the rehydration process was done by determining the indexes that take into account the water absorption capacity and solutes losses. The material dried by lyophilization show greater potential to rehydrate.

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(1997) yield monitor. However, coupled to a booming world sugar market, much of the **present** interest has been inspired by the recent availability to Australian cane farmers of grants which support adoption of farming methods which are perceived to reduce the impact of agriculture **on** the Great Barrier Reef; PA, and VRA in particular, is such a technology (BRAMLEY et al., 2008). There are two significant problems with this. First, the inaccuracy of the fertilizer delivery mechanisms used by Australian sugarcane growers (Dr Bernard Schroeder and John Panitz, BSES Ltd - pers. comm) raise serious questions about the merits of retro-fitting these with VRA controllers. Second, and arguably of more immediate importance, the **present** lack of a robust, commercially available sugarcane yield monitoring system in Australia (JENSEN et al., 2010) casts doubt **on** the basis for delineating management zones in sugar fields and thus, VRA. A major research effort which addresses these issues and the means by which sugar growers might adopt PA is presently underway. Nevertheless, auto-steer technology is being rapidly adopted in the Australian sugar industry and, as has been the case with grains (see above); this is expected to increase interest and adoption, which is otherwise lagging behind other major cropping industries.

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3.2 Permafrost distribution in historical simulation The simulated permafrost in JULES is shown in Fig. 7, along with observations from the Circum-Arctic map of permafrost and ground-ice conditions (Brown et al., 1998) (Sect. 2.4.6). The observed map shows areas with continuous, discontin- uous and sporadic permafrost and isolated patches. There is no equivalent of discontinuous permafrost in JULES be- cause each grid box has only a single soil column, so in or- der to compare the two maps we assume that a deeper active layer in JULES may correspond to discontinuous or sporadic permafrost. With this assumption, all the simulations match the observations fairly well in most areas. We can see that introducing the model **developments** brings in much more spatial variability in ALT, which generally matches with the patterns of continuous/discontinuous permafrost. The corre- lation between the ALT in JULES and the percentage cover of permafrost from (Brown et al., 1998) (100 % for continu- ous, 90 % for discontinuous, 50 % for sporadic and 10 % for isolated patches) is high, ranging between −0.37 and −0.51. However, there are still places where continuous per- mafrost is observed but JULES does not simulate permafrost. Figure 8 shows that in most of these areas, JULES simulates far too much snow, which will mean too much insulation in winter leading to soils that are too warm. This is particularly noticeable in north-east Canada and two areas in north-west Russia. In north-east Canada, however, it has been shown that the GlobSnow data set underestimates the SWE (Langlois et al., 2014), so the over-estimation in JULES may not be as large as Fig. 8 suggests. However, the permafrost in this re- gion is unstable to thawing (Thibault and Payette, 2009), so a small bias in the model could make the difference between simulating permafrost or not. For most of the remaining land surface, JULES slightly underestimates the SWE. Hancock et al. (2014) showed that JULES generally underestimates SWE when driven by reanalysis data sets.

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By identifying the orders of phase transition through the analytic continuation of the functional of the free energy of the Ehrenfest theory, we have developed a theory for studying the dependence of the local magnetic moment, M **on** the Fe – As layer sep- aration in the **third** **order** phase transition regime. We derived the Euler – Lagrange **equation** for studying the dynamics of the local magnetic moment, and tested our model with available experimental data.

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x (4) + ϕ( x) .. ... x + f (x, x, . x) + g(x, .. x) + h(x) = p(t, x, . x, . x, .. ... x ), (1.1) where ϕ, f, g, h and p are continuous functions which depend only **on** the argu- ments displayed. The dots indicate differentiation with respect to the independent variable t and all solutions considered are assumed real. The derivatives,

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Integral equations arise naturally from many applications in describing numer- ous real world problems, see, for instance, books by Agarwal et al. [1], Agarwal and O’Regan [2], Corduneanu [8], Deimling [13], O’Regan and Meehan [18] and the references therein. **On** the other hand, also quadratic integral equations have many useful applications in describing numerous events and problems of the real world. For example, quadratic integral equations are often applicable in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport and in the traffic theory. Especially, the so-called quadratic integral **equation** of Chandrasekher type can been countered very often in many applications; see for instance the book by Chandrasekher [7] and the research papers by Banas et al. [3, 4], Benchohra and Darwish [6], Darwish [9, 10, 11, 12], Hu et al. [15], Kelly [16], Leggett [17], Stuart [19] and the references therein. In [3] Banas et al. established the existence of monotonic solutions of a Volterra counter part of **equation** (1) by means of a technique associated with measure of noncompactness.

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As is widely known, since the concept of derivative was first developed, the ordinary differential equations have been used to model natural phenomena and they have provided a good approximation to determine its behavior. However, they involve functions and their derivatives that all evolve at the same time t. This is a feature that does not take into account the non-instantaneous nature of many phenomena. Additionally, phenomena with a delayed effect have been found. These phenomena are best modeled by a more general type of equations, the so-called delay differential equations (DDEs), making them one of the mathematical “tools” widely used in many applications ([1,2]). Such applications arise in a wide variety of fields as biophysics [3], biology [4], chemistry [5], climate [6] or even medicine [7], whenever it is essential to portray the non-instantaneous nature of the processes [8]. Within this framework, mathematical models for neural reflex mechanisms often take the form of a first-**order** DDE, that is

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tion of non linear terms into the evolution equation.. The region of.[r]

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