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    Bright soliton of the perturbed Schrödinger–Hirota equation with cubic–quintic–septic law of self-phase modulation in the presence of spatiotemporal dispersion
    (SPRINGER HEIDELBERGTIERGARTENSTRASSE 17, D-69121 HEIDELBERG, GERMANY, 2024) Özdemir, Neslihan; Altun, Selvi; Seçer, Aydın; Özışık, Müslüm; Bayram, Mustafa
    For the first time, we intend to scrutinize both the bright optical soliton solutions of the perturbed Schrödinger–Hirota equation with cubic–quintic–septic law having the spatiotemporal dispersion and the influences of the considered equation parameters on the soliton structure. The simple version of the new extended auxiliary equation method is utilized to carry out the aims. Taking the suitable complex wave transformation, the investigated equation becomes a nonlinear ordinary differential equation. Then, a system consisting of equations in polynomial structure utilizing the technique was able to produce. The bright optical solution is generated by utilizing the presented method. Finally, numerous projections of the bright soliton are indicated to explain the propagation of optical pulses in optic fibers. Furthermore, some depictions describing the effect of the model parameter were added.
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    Legendre Wavelet Operational Matrix Method for Solving Fractional Differential Equations in Some Special Conditions
    (VINCA INST NUCLEAR SCI, MIHAJLA PETROVICA-ALASA 12-14 VINCA, 11037 BELGRADE. POB 522, BELGRADE, 11001, SERBIA, 2019) Seçer, Aydın; Altun, Selvi; Bayram, Mustafa
    This paper proposes a new technique which rests upon Legendre wavelets for solving linear and non-linear forms of fractional order initial and boundary value problems. In some particular circumstances, a new operational matrix of fractional derivative is generated by utilizing some significant properties of wavelets and orthogonal polynomials. We approached the solution in a finite series with respect to Legendre wavelets and then by using these operational matrices, we reduced the fractional differential equations into a system of algebraic equations. Finally, the introduced technique is tested on several illustrative examples. The obtained results demonstrate that this technique is a very impressive and applicable mathematical tool for solving fractional differential equations.
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    Revealing optical soliton solutions of Schrödinger equation having parabolic law and anti-cubic law with weakly nonlocal nonlinearity
    (TAYLOR & FRANCIS LTD2-4 PARK SQUARE, MILTON PARK, ABINGDON OR14 4RN, OXON, ENGLAND, 2024) Özdemir, Neslihan; Altun, Selvi; Seçer, Aydın; Özışık, Müslüm; Bayram, Mustafa
    In this study, we purpose to ensure optical soliton solutions of the nonlinear Schrödinger equation having parabolic and anti-cubic (AC) laws with a weakly non-local nonlinearity by using the new Kudryashov method. As far as we know this model has not been presented and studied before. Furthermore, what differs this study from other studies is, not only obtains a variety of analytical solutions of the examined model but also substantiates the effects of the parabolic and anti-cubic laws with a weakly non-local nonlinearity on soliton behaviour, by choosing the particular soliton forms, which are dark, bright and W-like. Eventually, we depict some of the derived solutions in contour, 2D and 3D diagrams selecting the appropriate values of parameters by means of Matlab to demonstrate the importance of the given model. It is indicated that parabolic and AC parameters taking into consideration the weak non-local contribution have a very remarkable impact on the soliton structure, and the impact alters connected with the parameters and the soliton form. Besides, enabling and retaining the critical balance between the parameters and the soliton form and the interactive relation of the parameters with each other comprises major challenges.
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    Soliton solutions of Heisenberg spin chain equation with parabolic law nonlinearity
    (Springer, 2023) Altun, Selvi; Ozdemir, Neslihan; Ozisik, Muslum; Secer, Aydin; Bayram, Mustafa
    In this article, we presented and investigated a model of the (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation with parabolic law nonlinearity, for the first time. This study intends on two main elements. The first is to construct the parabolic law nonlinearity form of the HFSC model, and the second is to peruse the effect of this form on the soliton wave behavior. Two different analytical techniques, which are a subversion of the new extended auxiliary equation method and the improved generalized Kudryashov method (iGKM), are applicated to construct the soliton solutions of investigated HFSC form. With the help of these methods, bright, singular, and kink soliton forms have been obtained and their characteristics have been depicted by 2D and 3D diagrams. In accordance with our litterateur investigation, the subject of the study and the results obtained have not been reported before and we can say that the results we have attained will be a guide for researchers who want to study in this field.