Single and combined optical solitons,and conservation laws in (2+1)-dimensions with Kundu–Mukherjee–Naskar equation

In this work, the celebrated (2+1)-dimensional Kundu–Mukherjee–Naskar equation (KMNE) proposed to govern the soliton dynamics in (2+1)-dimensions along excited resonant wave guides that is doped with Erbium atoms is studied with the aid of ansatz approach and sine-Gordon expansion method (SGEM). The integration algorithms revealed both single and combined optical solitons of the model. These solitons are reported as bright, dark, combined dark-bright and singular solitons. The combined dark-bright and combined singular soliton solutions of the KMNE are to the best of our knowledge reported for the first time in this paper. These solutions supplements the existing ones in the literature. Additionally, we studied the conservation laws (Cls) of the equation by applying the multipliers approach and report the non-trivial fluxes associated with the equation. The physical structure of the obtained solutions are shown by graphic illustration in order to give a better understanding on the dynamics of optical solitons.     

Optical solitons for Radhakrishnan–Kundu–Lakshmanan equation with full nonlinearity

This paper retrieves bright, dark, singular and dark–singular combo dispersive optical solitons that stem from Radhakrishnan–Kundu–Lakshmanan model which is studied with full nonlinearity. These solitons are listed with constraint conditions on their parameters which serve as their existence criteria. Two strategically sound integration schemes have made soliton recovery successful.     

Optical solitons in birefringent fibers with Radhakrishnan–Kundu–Lakshmanan equation by a couple of strategically sound integration architectures

This paper addresses optical solitons in birefringent fibers modeled by Radhakrishnan–Kundu–Lakshmanan equation in coupled vector form. Bright, dark and singular solitons are recovered by trial and modified simple function principles.     

Optical solitons with Chen–Lee–Liu equation by Lie symmetry

This paper avails of classical Lie symmetry analysis to exhibit optical solitons to Chen–Lee–Liu model. By the aid of the proposed method, we secure symmetries that transform the model to a system of ordinary differential equations which are subsequently investigated by a number of methods to recover bright, dark and singular solitons.     

Optical solitons with complex Ginzburg–Landau equation having three nonlinear forms

This paper secures, dark, singular and dark–singular combo optical soliton solutions to complex Ginzburg–Landau equation that is considered with three nonlinear forms. Two forms of integration architectures provide these solutions.     

Interaction solutions for the(2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation in incompressible fluid

This paper aims to search for the solutions of the(2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation. Lump solutions, breather solutions, mixed solutions with solitons,and lump-breather solutions can be obtained from the N-soliton solution formula by using the long-wave limit approach and the conjugate complex method. We use both specific circumstances and general higher-order forms of the hybrid solutions as examples. With the help of maple software, we create density and 3D graphs...     

Duffin–Kemmer–Petiau Equation with a Hyperbolical Potential in (2+1) Dimensions for Spin-One Particles

We study the relativistic Duffin–Kemmer–Petiau equation in the presence of a hyperbolical potential in (1+2)-dimensional space–time for spin-one particles. To derive the energy eigenvalues and the corresponding eigenfunctions, we use the Nikiforov–Uvarov method after a Pekeris-type approximation is employed.     

Optical solitons in fiber Bragg gratings having Kerr law of refractive index with extended Kudryashov’s method and new extended auxiliary equation approach

This paper reveals optical solitons and other solutions to fiber Bragg gratings with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are indeed a technological marvel that supplements chromatic dispersion when its count runs low. The extended Kudryashov’s method and new extended auxiliary equation method have been implemented. Chirped and chirp–free bright, dark and singular solitons, with dispersive reflectivity, are presented.     

Optical solitons with complex Ginzburg–Landau equation for two nonlinear forms using F-expansion

This paper implements F-expansion scheme to obtain Jacobi’s elliptic function to complex Ginzburg–Landau equation with two nonlinear forms. In the limiting case of the modulus of ellipticity, bright and dark soliton solutions emerge.     

Exact solutions of nonlinear Schrodinger’s equation with dual power-law nonlinearity by extended trial equation method

In this paper, we acquire the soliton solutions of the nonlinear Schrodinger’s equation with dual power-law nonlinearity. Primarily, we use the extended trial equation method to find exact solutions of this equation. Then, we attain some exact solutions including soliton solutions, rational and elliptic function solutions of this equation using the extended trial equation method.     

Optical soliton perturbation with Kudryashov's equation by semi–inverse variational principle

The semi–inverse variational principle retrieved bright 1–soliton solution to the perturbed Kudryashov's equation. The perturbation terms appear with maximum intensity and additionally higher order dispersion effects are included. The parameter constraints for the solitons to exist are also enumerated.     

Optical solitons with differential group delay for Kudryashov’s model by the auxiliary equation mapping method

In this paper, the auxiliary equation mapping method is employed to extract optical solitons and other solutions for special cases of Kudryashov’s model in birefringent fibers that is studied without the effect of four wave mixing effects. Bright, dark and singular solitons and other solutions emerge from the auxiliary equation mapping method.     

Soliton structures in the (1+1)-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics

In this paper, the(1+1)-dimensional variable-coefficient complex Ginzburg–Landau(CGL) equation with a paritytime(PT) symmetric potential U(x) is investigated. Although the CGL equations with a PT-symmetric potential are less reported analytically, the analytic solutions for the CGL equation are obtained with the bilinear method in this paper. Via the derived solutions, some soliton structures are presented with corresponding parameters, and the influences of them are analyzed and studied. The single-soliton structure is numerically verified, and its stability is analyzed against additive and multiplicative noises. In particular, we study the soliton dynamics under the impact of the PT-symmetric potential. Results show that the PT-symmetric potential plays an important role for obtaining soliton structures in ultrafast optics, and we can design fiber lasers and all-optical switches depending on the different amplitudes of soliton-like structures.     

Localized structures for (2+1)-dimensional Boiti–Leon–Pempinelli equation

It is shown that Painlevé integrability of (2+1)-dimensional Boiti–Leon–Pempinelli equation is easy to be verified using the standard Weiss–Tabor–Carnevale (WTC) approach after introducing the Kruskal’s simplification. Furthermore, by employing a singular manifold method based on Painlevé truncation, variable separation solutions are obtained explicitly in terms of two arbitrary functions. The two arbitrary functions provide us a way to study some interesting localized structures. The choice of rational functions leads to the rogue wave structure of Boiti–Leon–Pempinelli equation. In addition, for the other choices, it is observed that two solitons may evolve into breather after interaction. Also, the interaction between two kink compactons is investigated.     

Optical solitons and periodic solutions of the (2 + 1)-dimensional nonlinear Schrödinger's equation

In this Letter, we consider the $(2+1)$-dimensional nonlinear Schrödinger's equation. With the aid of the Jacobian elliptic equation, we derive the exact bright soliton, dark soliton, singular soliton and periodic solutions of this equation expressed in terms of trigonometric functions, hyperbolic functions and Jacobian elliptic functions, respectively. Finally, for certain parametric values, we plot three dimensional graphics of modulus, real and imaginary parts of some solutions, which can help one better understand their dynamical behavior via their graphics analysis.     

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg–de Vries equation

Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.     

A transformed rational function method for (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation

A direct method, called the transformed rational function method, is used to construct more types of exact solutions of nonlinear partial differential equations by introducing new and more general rational functions. To illustrate the validity and advantages of the introduced general rational functions, the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama (YTSF) equation is considered and new travelling wave solutions are obtained in a uniform way. Some of the obtained solutions, namely exponential function solutions, hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions and rational solutions, contain an explicit linear function of the independent variables involved in the potential YTSF equation. It is shown that the transformed rational function method provides more powerful mathematical tool for solving nonlinear partial differential equations.     

Optical color image hiding scheme by using Gerchberg–Saxton algorithm in fractional Fourier domain

We proposed an optical color image hiding algorithm based on Gerchberg–Saxton retrieval algorithm in fractional Fourier domain. The RGB components of the color image are converted into a scrambled image by using 3D Arnold transform before the hiding operation simultaneously and these changed images are regarded as the amplitude of fractional Fourier spectrum. Subsequently the unknown phase functions in fractional Fourier domain are calculated by the retrieval algorithm, in which the host RBG components are the part of amplitude of the input functions. The 3D Arnold transform is performed with different parameters to enhance the security of the hiding and extracting algorithm. Some numerical simulations are made to test the validity and capability of the proposed color hiding encryption algorithm.     

Chaotic solutions of (2+1)-dimensional Broek Kaup equation with variable coefficients

In this paper,an improved projective approach is used to obtain the variable separation solutions with two arbitrary functions of the (2+1)-dimensional Broek-Kaup equation with variable coefficients (VCBK). Based on the derived solitary wave solution and using a known chaotic system,some novel chaotic solutions are investigated.     

Two-Body Scattering in(1+1) Dimensions by a Semi-relativistic Formalism and a Hulthén Interaction Potential

Scattering solutions of two-body Spinless Salpeter Equation(SSE) are investigated in the center of mass frame with a repulsive, symmetric Hulth′en potential in one spatial dimension. Transmission and reflection coefficients are calculated and discussed.