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There is a very wide application of mathematics in communication theory, signal processing and networking. A network is formed by nodes communicating with each other. Graph theory plays a vital role in the area of networking research. A topological index is a numeric quantity obtained from a graph structure which is invariant under graph isomorphism. In this study, we obtained a generalized degree-based topological index, called -Zagreb index of some derived networks such as Sierpinski network, butterfly network, Benes network and mesh-derived network and hence consider some particular cases. Council of Scientific & Industrial Research (CSIR) - India 09/973(0016)/2017-EMR-I

Optical soliton pulses or molecules with the Sasa-Satsuma equation have been worked by modified simple equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

The Kundu–Mukherjee–Naskar model in birefringent fibers is examined to uncover quite important optical solitons in (2+1)-dimensions of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Triki-Biswas equation that is a new model for soliton dynamics of sub pico-second pulses in a monomodal optical fiber with non-Kerr dispersion is considered to get optical solitos of its in this paper. Optical bright and singular solitons are given by trial equation approach. Also, singular periodic solutions are located.

The Chen–Lee–Liu equation in birefringent fibers is examined to uncover quite important optical solitons of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Biswas–Arshed model which is a new model for soliton transmission through optical fibers is employed to obtain optical solitons of its in this work. The trial equation technique is used to catch out both highly important solitons and singular periodic solutions.

The Biswas-Arshed model in birefringent fibers which is first written in two component form for vector solitons is employed to obtain optical solitons of its in this work. The trial equation technique is used to catch out both highly important solitons and singular periodic solutions.

The inverse problem for integer-valued topological indices is about the existence of a graph having its index value equal to a given integer. We solve this problem for the first and second Zagreb indices, and present analogous results also for the forgotten and hyper-Zagreb index. The first Zagreb index of connected graphs can take any even positive integer value, except 4 and 8. The same is true if one restricts to trees or to molecular graphs. The second Zagreb index of connected graphs can take any positive integer value, except 2, 3, 5, 6, 7, 10, 11, 13, 15 and 17. The same is true if one restricts to trees or to molecular graphs.

This paper examines soliton dynamics of sub pico-second pulses modeled by Kaup–Newell equation. The modified simple equation scheme and trial equation approach retrieves dark, bright as well as singular solitons to the model. The constraint relations guarantee the existence of such solitons.

This paper studies differential group delay in optical fibers with coupled Fokas-Lenells equation. Two integration schemes are adopted. They are trial equation method and modified simple equation scheme. Bright, dark and singular soliton solutions are retrieved together with their respective existence criteria that are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains bright, dark and singular dispersive optical soliton solutions to Schrodinger-Hirota equation by the aid of trial equation method. Both Kerr and power laws of nonlinearity are studied. Singular periodic solutions are also obtained as a byproduct of this scheme. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

The modified simple equation scheme is employed to secure dark and singular optical soliton solution to the perturbed Gerdjikov-Ivanov equation that models pulse dynamics in optical fibers and PCF. The solvability conditions for this algorithm is also presented. National Science Foundation for Young Scientists of Wuhan Donghu University (2017dhzk001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains optical soliton solution to perturbed Gerdjikov-Ivanov equation by trial equation approach. Bright, dark and singular soliton solutions are derived. Additional solutions such as singular periodic solutions also emerge of this integration scheme. National Science Foundation for Young Scientists of Wuhan Donghu University (2017dhzk001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains optical soliton solutions to the Kundu Eckhaus equation with general coefficients. The Riccati Bernoulli's sub-ODE method as well as Kudryashov's scheme are employed to obtain soliton solutions to the model. This generalized earlier reported result of the model with specific coefficients. Subsequently, the multiplier approach was utilized to secure a conserved quantity with the bright soliton solution that was reported earlier. Young Foundation of Wuhan Donghu University (2017DHKZ001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.

In this study, we consider three model equations of shallow water waves. Shallow water equations model the propagation of strongly nonlinear waves up to breaking and run-up in nearshore zones. We perform multiple exp-function method which is known as a generalization of Hirota's perturbation scheme. We yield one-, two-, and three-wave solutions. The obtained solutions can be used as benchmarks for numerical solutions of the underlying equations. Faculty Research Committee of FAST, North-West University, Mafikeng Campus, SouthAfrica

Optical soliton pulses or molecules with the Sasa–Satsuma equation have been worked by trial equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

The Kundu–Mukherjee–Naskar model is examined to uncover quite important optical solitons in (2 + 1)-dimensions of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Biswas-Arshed model which is a new model for soliton transmission through optical fibers is taken into account for the purpose of acquiring very valuable optical solitons of its in this paper. The modified simple equation methodology is utilized for obtaining both highly important solitons and singular periodic solutions.

This paper implements two efficient integration algorithms to retrieve soliton solutions that appear with weak nonlocal nonlinearity. These are trial equation method and modified simple equation scheme. Bright, dark and singular soliton solutions are recovered and their existence criteria are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

The trial equation method is applied to obtain soliton solutions to Lakshmanan–Porsezian–Daniel model in optical fibers, PCF and metamaterials. This integration procedure is implemented into the model with three forms of nonlinearity. Bright, dark and singular soliton solutions are recovered with conditions that guarantee their existence.

This paper retrieves soliton solutions to the perturbed resonant nonlinear Schrodinger's equation with time-dependent coefficients using the modified simple equation method. The four types of exotic non-Kerr laws studied in this paper are quadratic-cubic law, anti cubic law, cubic-quintic-septic law and triple-power law. These soliton solutions appear with constraint conditions that guarantee their existence. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper retrieves optical soliton solution to the perturbed complex Ginzburg-Landau equation that is studied with nine different forms of nonlinearity. The trial solutions approach is the integration algorithm adopted in this paper. The perturbation terms appear with full nonlinearity to get a taste of generalized setting. Bright, dark and singular soliton solutions are obtained. The existence criteria of such solitons are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper secures bright, dark and singualr soliton solutions in birefringent fibers. Both Kerr law and parabolic law nonlinearity are studied. The trial equation method successfully retrieves these solitons with constraint relations that assure their existence. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

The modified simple equation method applied to perturbed complex Ginzburg-Landau equation gives dark and singular solutions to the model. The perturbation terms appear with full nonlinearity. There are eight nonlinear forms studied in this paper. These solitons appear with constraint conditions that gurantee its existence and these are also presented. Chutian Scholar Program of Hubei Government in China Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

Let Σ g denote a closed orientable surface of genus g ≥ 2 . We consider a certain infinite family of Σ g -bundles over circle whose monodromies are taken from some collection of pseudo-Anosov diffeomorphisms. We show the existence of tight contact structure on every closed 3-manifold obtained via rational r -surgery along a section of any member of the family whenever r ≠ 2 g − 1 . Combining with Thurston's hyperbolic Dehn surgery theorem, we obtain infinitely many hyperbolic closed 3-manifolds admitting tight contact structures.

In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized tractrices in Euclidean (n + 1)-space En+1. Further, we introduce some kind of generalized rotational surfaces in Euclidean spaces E-3 and E-4, respectively. We have also obtained some basic properties of generalized rotational surfaces in E-4 and some results of their curvatures. Finally, we give some examples of generalized Beltrami surfaces in E-3 and E-4, respectively.

This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly K\"ahler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface of nearly K\"ahler manifold. It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.

In this paper Lie symmetry analysis of the seventh-order time fractional Sawada–Kotera–Ito (FSKI) equation with Riemann–Liouville derivative is performed. Using the Lie point symmetries of FSKI equation, it is shown that it can be transformed into a nonlinear ordinary differential equation of fractional order with a new dependent variable. In the reduced equation the derivative is in Erdelyi–Kober sense. Furthermore, adapting the Ibragimov’s nonlocal conservation method to time fractional partial differential equations, we obtain conservation laws of the underlying equation. In addition, we construct some exact travelling wave solutions for the FSKI equation using the sub-equation method. Keywords: Fractional Sawada–Kotera–Ito equation, Lie symmetry, Riemann–Liouville fractional derivative, Conservation laws, Exact solutions

In this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov’s new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer–Chree (PC) equation and the Kaup–Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.

In this paper we study the conservation laws of modified Korteweg-de Vries-Kadomtsev Petviashvili equation (mKdV-KP). As the considered equation is of evolution type, no recourse to a Lagrangian formulation is made. However, we show that using the partial Lagrangian approach and the multiplier method one can obtain a number of local and nonlocal conservation laws for underlying equation.

We applied the multiple exp-function scheme to the (2+1)-dimensional Sawada-Kotera (SK) equation and (3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota's perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.