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The Complex Ginzburg-Landau equation (CGLE) has been suggested and investigated as a model for turbulent dynamics in nonlinear partial differential equations. It is a particularly intriguing model in this aspect since it is a dissipative form of the nonlinear Schrödinger equation. The model is studied with quadratic-cubic law nonlinear fiber by using ϕ6–model expansion method, bright, dark, dark-singular, singular and dark-bright optical soliton solutions are obtained. The findings of this study might assist in comprehending some of the physical implications of various nonlinear physics models. The hyperbolic sine, for example, appears in the calculation of the Roche limit and gravitational potential of a cylinder, while the hyperbolic cosine function is the shape of a hanging cable (the so-called catenary). Finally, there are some discussions regarding new complex solutions, it is explored by giving physical meaning to the constants found in traveling wave solutions, which are both physically and mathematically significant. The three-dimensional, two-dimensional simulations and contour plots are used to enhance these discussions.

This work investigates the complex Ginzburg–Landau equation (CGLE) with parabolic law in nonlinear optics, this form of nonlinearity may also be seen in fiber optics. It is referred to as the fifth-order susceptibility, which is predominantly present in a transparent glass with intense femtosecond pulses at 620nmI. The ϕ6-model expansion approach is used to find dark, singular, periodic, and combined optical soliton solutions to the model. The results presented in this study are intended to improve the CGLE’s nonlinear dynamical characteristics. These solitons are significant resources in physics and telecommunications engineering. They led to several quick follow-up investigations. The hyperbolic sine, for example, appears in the calculation of the Roche limit and gravitational potential of a cylinder, while the hyperbolic cotangent appears in the Langevin function for magnetic polarization. Some of the obtained solutions’ 2-, 3-dimensional, and contour plots are shown.

In this study, the (1/G') -expansion method was implemented to solve the Benney–Luke (BL) equation. Exact solutions of the BL equation were obtained via this method. The solutions obtained from the BL equation were in hyperbolic form. 3D, 2D and contour graphs of obtained solutions are presented. Results show that the (1/G') -expansion method provides an efficient and straightforward mathematical instrument for finding solutions of nonlinear evolution equations (NLEEs)., Bu çalışmada, Benney-Luke (BL) denklemini çözmek için (1/ G') -açılım yöntemi uygulanmıştır. BL denkleminin tam çözümleri bu yöntem ile elde edilmektedir. BL denkleminden elde edilen çözümler hiperbolik formdadır. Elde edilen çözümlerin 3 boyutlu, 2 boyutlu ve kontur grafikleri sunulmaktadır. Sonuçlar, (1/ G') - açılım yönteminin doğrusal olmayan evrim denklemlerinin çözümlerini bulmak için etkili ve basit bir matematiksel enstrüman olduğu gösterilmiştir.

In this paper, for a given curve in the Euclidean 3-space $\mathbb{R}^{3}$ we introduce new invariants such as arc-length, curvature and torsion withfractional-order and provide certain relations between these and the standart invariants. We obtain the Frenet-Serret formulas in $\mathbb{R}^{3}$and then construct the ways of determining a curve in $\mathbb{R}^{2}$ and $% \mathbb{R}^{3}$ in terms of the new invariants. Several examples are alsogiven by figures.

Bu çalışmanın genel amacı (1/G') -açılım yöntemi kullanılarak Düzenli Uzun Dalga (RWL) denklemi için yürüyen dalga çözümlerini elde etmektir Elde Edilen çözümlerde sabitlere özel değerler verilerek 3 boyutlu, 2 boyutlu ve kontur grafikleri sunulmuştur. Bu grafikler Düzenli Uzun Dalga denkleminin özel bir çözümüdür ve denklemin durağan bir dalgasını temsil etmektedir. Bu makalede sunulan çözümleri ve grafikleri bulmak için bilgisayar paket programı kullanılmaktadır., The main goal of this study is to obtain the traveling wave solutions for Regularized Long Wave (RLW) equation by using (1/G') -expansion method. By giving special values to the constants in the solutions obtained, 3D, 2D, and contour graphics are presented. These graphics are a special solution of the (RLW) equation, and they represent a stationary wave of the equation. A computer package program is used to find the solutions and graphics presented in this article.

The main goal of this study is obtaining analytical solutions for (3+1)-dimensional Jimbo-Miwa Equation which the second equation in the well-known KP hierarchy of integrable systems. For the (3+1DJM) equation, hyperbolic, trigonometric, complex trigonometric and rational traveling wave solutions have been constructed by applying the (G'/G,1/G)-expansion method. Then, real and imaginary graphics are presented by giving special values to the constants in the solutions obtained. These graphics are a special solution of the (3+1DJM) equation and represent a stationary wave of the equation. The (G'/G,1/G)-expansion method is an effective and powerful method for solving nonlinear evolution equations (NLEEs). Ready computer package program is used to obtain the solutions and graphics presented in this study.

In the current study, analytical solutions are constructed by applying (1/G') -expansion method to the Kolmogorov–Petrovskii–Piskunov (KPP) equation. Hyperbolic type exact solutions of the KPP equation are presented with the successfully applied method. 3D, 2D and contour graphics are presented by giving special values to the parameters in the solutions obtained. This article explores the applicability and effectiveness of this method on nonlinear evolution equations (NLEEs)., Bu çalışmada, Kolmogorov – Petrovskii –Piskunov (KPP) denkleminin analitik çözümleri (1/G')-açılım yöntemi uygulanarak elde edilmiştir. Başarılı bir şekilde uygulanan yöntem ile KPP denkleminin hiperbolik tipte tam çözümleri sunulmuştur. Elde edilen çözümlerdeki parametrelere özel değerler verilerek 3 boyutlu, 2 boyutlu ve kontur grafikleri sunulmuştur. Bu makalede, bu yöntemin doğrusal olmayan evrim denklemleri (NLEE'ler) üzerindeki uygulanabilirliği ve etkinliği araştırılmaktadır.

The purpose of this article is obtaining the exact solutions for (3+1)-dimensional Jimbo-Miwa Equation (3+1DJME). The (1/G')-expansion method which is an effective method in solving nonlinear evolution equations (NLEEs) is used. Then, 3D, contour and 2D graphics are presented by giving special values to the constants in the solutions obtained. These graphics are a special solution of the (3+1DJME) and represent a stationary wave of the equation. Ready computer package program is used to obtain the solutions and graphics presented in this study., Bu makalenin amacı (3+1) boyutlu Jimbo-Miwa denklemi için tam çözümler elde etmektir. Lineer olmayan evrim denklemlerinin çözümünde etkili bir yöntem olan (1/G')-açılım yöntemi kullanılmıştır. Daha sonra elde edilen çözümlerdeki sabitlere özel degerler verilerek 3 boyutlu, kontur ve 2 boyutlu grafikler sunulmuştur. Bu grafikler (3 + 1) boyutlu Jimbo-Miwa denkleminin özel bir çözümü olup ve denklemin duragan bir dalgasını temsil etmektedir. Bu çalışmada sunulan çözümler ve grafiklerin elde edilişinde hazır bilgisayar paket programı kullanılmaktadır.