Showing papers in "Optical and Quantum Electronics in 2018"
TL;DR: In this paper, the generalized projective Riccati equation method is proposed to establish exact solutions for generalized form of the reaction Duffing model in fractional sense namely, Khalil's derivative.
Abstract: The generalized projective Riccati equation method is proposed to establish exact solutions for generalized form of the reaction Duffing model in fractional sense namely, Khalil’s derivative. The compatible traveling wave transform converts the governing equation to a non linear ODE. The predicted solution is a series of two new variables that solve a particular ODE system. Coefficients of terms in the series are calculated by solving an algebraic system that comes into existence by substitution of the predicted solution into the ODE which is the result of the wave transformation of the governing equation. Returning original variables give exact solutions to the governing equation in various forms.
102 citations
TL;DR: In this article, the authors investigated the Nizhnik-Novikov-Veselov and the Drinfel-d-Sokolov systems by using the extended sinh-Gordon equation expansion method.
Abstract: This study investigates the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems by using the extended sinh-Gordon equation expansion method. The Nizhnik-Novikov-Veselov system is an isotropic Lax extension of the one-dimensional Korteweg-de Vries equation and the Drinfel’d-Sokolov system is a nonlinear model that possesses some special forms of Lax pairs. Topological kink-type, topological, non-topological, compound topological-non-topological bell-type and singular soliton solutions are successfully constructed. Under the choice of suitable values of parameters, the 2D, 3D and contour graphs to some of the obtained solutions are plotted.
91 citations
TL;DR: In this paper, three nonlinear resonant rings inside two dimensional photonic crystals structure for realizing all optical halfadder gate were created by adding some nonlinear rods to the photonic crystal based ring resonator structure.
Abstract: In this paper we used three nonlinear resonant rings inside two dimensional photonic crystals structure for realizing all optical half-adder gate. The nonlinear resonant rings were created by adding some nonlinear rods to the photonic crystal based ring resonator structure. The proposed structure has two input and two output ports. The simulations show that when both input are off, both output ports will be off. When one input is ON, port S will be on and when both inputs are on port C will be on. S and C serve as the sum and carry ports of the optical half-adder structure. The maximum rise and fall times of the proposed structure are about 1.5 and 1 ps.
87 citations
TL;DR: In this paper, the properties of pristine and vacancy defect monolayer tungsten diselenide (WSe2) have been investigated by first principles calculations, and it was shown that the defects in WSe2 can increase the work function of pristine monolayers.
Abstract: Optoelectronic properties of pristine and vacancy defect monolayer tungsten diselenide (WSe2) have been investigated by the first principles calculations. The results predicate that Se defect monolayer WSe2 is direct semiconductor whereas the W defect monolayer WSe2 is metallic. The Se defect can decrease the work function for monolayer WSe2, however, the W defect can increase the work function for monolayer WSe2. The absorption edge for defect monolayer WSe2 occurs obviously red-shift, and the energy loss of electron transmitting in defect monolayer WSe2 is faster than pristine monolayer WSe2. The work gives a theoretical guidance for the fabrication of monolayer WSe2 optoelectronic nanodevices.
87 citations
TL;DR: In this paper, the unstable nonlinear Schrodinger's equation is studied, which points out the time evolution of disturbances in marginally stable or unstable media and a wide variety of new explicit exact solutions are successfully derived, proving the excellent performance of the schemes.
Abstract: Under investigation in the present work is to study the unstable nonlinear Schrodinger’s equation which points out the time evolution of disturbances in marginally stable or unstable media The exp
a
and hyperbolic function methods are adopted to carry out this target in a straightforward way A wide variety of new explicit exact solutions are successfully derived, proving the excellent performance of the schemes
80 citations
TL;DR: In this article, the coupled Schrodinger-Boussinesq equation with variable-coefficients using the unified method was investigated and new nonautonomous complex wave solutions were obtained and classified into two categories, namely polynomial function and rational function solutions.
Abstract: This paper investigates the coupled Schrodinger–Boussinesq equation with variable-coefficients using the unified method. New nonautonomous complex wave solutions are obtained and classified into two categories, namely polynomial function and rational function solutions. For the polynomial functions emerge the complex solitary, complex soliton and complex elliptic wave solutions, while for the rational function are observed complex periodic rational and complex hyperbolic rational wave solutions. The physical insight and the dynamical behavior of the solutions describing the wave propagation in laser or plasma physics are discussed and analysed for different choices of the arbitrary functions in the solutions.
70 citations
TL;DR: In this paper, the authors investigate for new exact solutions of higher order nonlinear Schrodinger's equation, which allows to carry out the solution process of nonlinear wave equations more thoroughly and conveniently by computer algebra systems such as Mathematica.
Abstract: Nonlinear Schrodinger equation (NLSE) is now one of the prominent of modern physics, mathematics and chemistry. Over these fields, the NLSE is also applied in new emerging fields such as quantum information and econophysics. In this paper we investigate for new exact solutions of higher order nonlinear Schrodinger’s equation. This method allows to carry out the solution process of nonlinear wave equations more thoroughly and conveniently by computer algebra systems such as the Maple and Mathematica. In addition to providing a different way of solving the Schrodinger equation for such systems, the simplicity of the algorithm renders it a great pedagogical value.
67 citations
TL;DR: In this paper, an ultra-compact structure is presented for realization of all-optical NOT and XOR logic gates which can be compatible with silicon technology, and the lattice constant and the radius of the rods are selected in such a way in order to operate the logic gates at 1550nm.
Abstract: In this paper, an ultra-compact structure is presented for realization of all-optical NOT and XOR logic gates which can be compatible with silicon technology. Logic gates are based on two-dimensional photonic crystals, and the lattice constant and the radius of the rods are selected in such a way in order to operate the logic gates at 1550 nm. The proposed structure consists of three waveguides which are connected to each other using a T-shaped junction. This structure is optimized by two nano-resonators and has also two input ports and one output port. For our numerical studies, the plane-wave expansion and finite-difference time-domain methods have been used. The contrast ratios for the proposed all optical NOT and XOR logic gates are respectively 43.40 and 43.38 dB. The response time of the logic gates is 0.37 ps, which in turn creates a data transmission rate of 3.15 Tb/s. Our studies have shown that the NOT-designed logic gate is suitable for the use in optical integrated circuits.
54 citations
TL;DR: In this article, the authors studied the time fractional generalized Burgers-Huxley equation with Riemann-Liouville derivative via Lie symmetry analysis and power series expansion method.
Abstract: In this work, we study the time fractional generalized Burgers–Huxley equation with Riemann–Liouville derivative via Lie symmetry analysis and power series expansion method We transform the governing equation to nonlinear ordinary differential equation of fractional order using its Lie point symmetries In the reduced equation, the derivative is in Erdelyi–Kober sense We apply power series technique to derive explicit solutions for the reduced equation The convergence of the obtained power series solutions are also derived Some interesting Figures for the obtained solutions are presented
48 citations
TL;DR: In this paper, the unstable nonlinear Schrodinger equation (UNLSE) was analyzed by modified extended direct algebraic method, which describes the disturbances in time evolution of marginally stable or unstable media.
Abstract: In this article, we study the unstable nonlinear Schrodinger equation (UNLSE). Analytically by modified extended direct algebraic method, which describes the disturbances in time evolution of marginally stable or unstable media. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave and elliptic function solutions of UNLSE. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomenas of this unstable equation. Moreover, we also present the formation conditions of the bright and dark solitons of UNLSE. The obtained results and computational work shows the power and effectiveness of this method. Many other such types of nonlinear evolution equations arising in engineering, applied sciences and nonlinear optics can also be solved by this method.
46 citations
TL;DR: In this paper, the authors proposed the first integral method (FIM) and the improved version of ITEM for solving the density-dependent conformable fractional diffusion reaction equation (CFDRE) which is commonly applied in mathematical biology.
Abstract: In this paper, we propose the first integral method (FIM) and the improved $$\text{tan}\left( {\frac{1}{2}{\varphi }\left(\upxi \right)} \right)$$
-expansion method (ITEM) for solving the density-dependent conformable fractional diffusion–reaction equation (CFDRE) which is commonly applied in mathematical biology. We received many new exact soliton solutions for the density-dependent CFDRE which are expressed by exponential function, rational function and hyperbolic function forms. The results show that FIM and ITEM are powerful mathematical tools and efficient techniques for solving the fractional nonlinear partial differential equations.
TL;DR: In this article, the design and performance enhancement of the power penalty in a dense wavelength division multiplexing based on free space optical communication (FSOC) link using digital pulse position modulation (DPPM) and on-off keying (OOK) modulation is presented.
Abstract: This paper presents the design and performance enhancement of the power penalty (PP) in a dense wavelength division multiplexing based on free space optical communication (FSOC) link using digital pulse position modulation (DPPM) and on–off keying (OOK) modulation. Such a system has a high performance, low cost, robust and power efficient, reliable, excessive flexibility, and higher data rate for access networks. The system performance is evaluated for an 8-channel wavelength-division-multiplexing for hybrid fiber FSOC system at 2.5 Gbps on widely accepted modulation schemes under various atmospheric turbulence (AT) regimes conditions. The performance of system is introduced in terms of PP, bit-error rate (BER), transmission distance and the average received optical power. The numerical results shows that the improvement of the PP using DPPM modulation of 0.2–3.0 dB for weak turbulence (WT) regimes for BER of 10−6 and above 20, 25 dB for strong turbulence (ST) regimes are reported for BER of 10−6 and 10−9, as respectively (depending on the AT level). Further, we develop of improvement the PP caused by multiple-access interference about 6.686 dB which is predicted for target BER of 10−9 in WT and 1 dB at target BER of 10−6 in ST when the 8 user are active on the system of optical network units. Additionally, the optical power budget and margin losses of a system are calculated with different link length. The proposed approach of DPPM merges superiority with higher enhancement of PP about 0.8 dB for BER equal 10−9 at FSO link length lfso = 2000 m compared to OOK at 1 dB for WT. An improvement of 2 dB is observed using the DPPM scheme over an OOK due to capability of detect pulses under background noise conditions with increased receiver sensitivity.
TL;DR: In this paper, the authors reveal bright, dark and singular soliton solutions to the perturbed Gerdjikov-Ivanov equation by the aid of extended Kudryashov's method.
Abstract: This paper reveals bright, dark and singular soliton solutions to the perturbed Gerdjikov–Ivanov equation by the aid of extended Kudryashov’s method. The nonlinear terms appear with full nonlinearity in order to give a generalized flavor to the model. As a byproduct of this scheme, plane waves and singular periodic solutions fall out and these solutions are listed as well.
TL;DR: In this paper, two different structure dimensions are used to analyze the sensing characteristics of urine, including square ring joined to a simple waveguide with rods in air configuration, and two-dimensional photonic crystals based on air hole in slab with elliptical resonant cavity in the middle of a photonic crystal waveguide.
Abstract: In this paper, we suggest two-dimensional photonic crystal based biosensors for measurement of urea concentration in urine by means of refractive index detecting. In case of variation of urea concentration in urine, both the output peak intensity and the resonant peak center wavelength will shift. Two different structure dimensions are used to analyze the sensing characteristics of urine. The first sensor consists of a novel square ring joined to a simple waveguide with rods in air configuration. The second sensor is schemed by use of two-dimensional photonic crystals based on air hole in slab with elliptical resonant cavity in the middle of a photonic crystal waveguide. To realize sensing in both cases, we fill air area by urine sample. A high sensitivity is observed in small structures. In addition, we demonstrated a high quality factor, which is superior to those reported in recently published work discussing urine components based on photonic crystal, with small size sensors and fast response times.
TL;DR: In this article, a new nanolithography technique in which sample rotation is incorporated into zeroth-order waveguide mode interference lithography is proposed, where a 325-nm laser was used to excite the waveguide modes, which were loaded by an asymmetric metal-cladding dielectric waveguide structure.
Abstract: A new nanolithography technique in which sample rotation is incorporated into zeroth-order waveguide mode interference lithography is proposed in this report. A 325-nm laser was used to excite zeroth-order waveguide modes, which were loaded by an asymmetric metal-cladding dielectric waveguide structure. The optical field intensity distribution of zeroth-order waveguide modes interference is numerically simulated using the finite element method. The lithography sample consisted of a glass substrate, Al film, and photoresist film, and the rotation operation on the sample is expressed in coordinate matrix transformation. Various subwavelength structures, such as two-dimensional square lattices, two-dimensional hexagonal closed-packed lattices,and circular gratings, were obtained through double, triple, and continuous exposure. These subwavelength structures with different sizes can be produced by changing the thickness of the photoresist. The subwavelength structures simulated with various shapes and sizes can be applied to the field of nano-optics. The proposed technique provides a flexible and promising approach for interference nanolithography because of its simplicity and low cost.
TL;DR: In this article, a new integrable nonlinear Schrodinger type equation in (2+1)-dimensions is investigated, which describes the spin dynamics of Heisenberg ferromagnetic spin chains.
Abstract: There are various types of fractional derivatives in literature e.g. Caputo, Riemann–Liouville, conformable and so on. In the present paper, a new integrable nonlinear Schrodinger type equation in (2+1)-dimensions is investigated, which describes the spin dynamics of Heisenberg ferromagnetic spin chains. Some bright and dark solitons and other exact solutions of this equation with conformable time fractional derivative are reported. Then, fractionalize effects of this model are graphically investigated. Moreover, the simplest equation method and Nucci’s method are utilized for differential equations with conformable fractional derivatives.
TL;DR: In this paper, the authors adopt the extended trial equation method to handle optical nonlinear negative-index materials that come with anti-cubic nonlinearity and obtain bright and singular solitons only.
Abstract: This paper adopts the extended trial equation method to handle optical nonlinear negative-index materials that come with anti-cubic nonlinearity. In presence of Hamiltonian perturbation terms, soliton solutions are obtained. These are bright and singular solitons only. However, some additional wave solutions are also emerged.
TL;DR: In this article, a D-shaped photonic crystal fiber with a nanoscale gold belt is used to produce surface plasmon resonance, which can be applied to the fields of chemical and biological sensing field.
Abstract: A novel refractive index sensor based on a D-shaped photonic crystal fiber with a nanoscale gold belt is proposed. The nanoscale gold belt is deposited on the flat surface to produce surface plasmon resonance. Numerical results show that the refractive index sensor has a broad measurement range which is from 1.2 to 1.4 and the maximum sensitivity can reach to 3751.5 nm/RIU. The refractive index resolution of the sensor is under 1 × 10−5 RIU in the whole measuring range. A high adjusted R-squared, up to 0.99002, is acquired from the fitting curve of the resonance wavelength. The sensor is sensitive to the variation of the refractive index of analytes and can be applied to the fields of chemical and biological sensing field.
TL;DR: In this paper, soliton solutions and stability analysis for conformable Cahn-Allen and conformable Zoomeron equations are presented, where the sine-Gordon method is used to carry out the soliton solution for these equations.
Abstract: This research presents soliton solutions and stability analysis to some conformable nonlinear partial differential equations (CNPDEs). The CNPDEs equations in this paper are conformable Cahn–Allen and conformable Zoomeron equations. The powerful sine-Gordon method is used to carry out the soliton solutions for these equations. The aspects of stability analysis for the considered equations is investigated using the linear stability technique. The sine-Gordon method proves to be efficient and effective for the extraction of soliton solutions for different types of CNPDEs.
TL;DR: In this article, exact analytical solutions for the time fractional variant bussinesq equations are constructed in the sense of the newly devised fractional derivative called conformable fractional derivatives.
Abstract: In this work, exact analytical solutions for the time fractional variant bussinesq equations are constructed in the sense of the newly devised fractional derivative called conformable fractional derivative. Using wave transformation, we converted the problem under consideration to an ordinary differential equation and then employed the modified extended tanh expansion method for hyperbolic function solutions. The Mathematica software is used throughout for the solution of the system of algebraic equations obtained along the way and also for the graphical illustrations, respectively.
TL;DR: In this paper, the modified Kudryashov method was successfully employed and acquired some new exact solitary wave solutions in terms of exponential based functions with fractional version, which can help us understand the variation of solitary waves in the field of oceanography.
Abstract: The present study emphasis to look for new closed form exact solitary wave solutions for the variety of fractional Boussinesq-like equations using the modified Kudryashov method with the help of symbolic computation. As a consequence, the modified Kudryashov method is successfully employed and acquired some new exact solitary wave solutions in terms of exponential based functions with fractional version. All solutions have been verified back into its corresponding equation with the aid of Maple package program. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed method is robust and efficient than other methods and the obtained solutions in this paper can help us to understand the variation of solitary waves in the field of oceanography.
TL;DR: In this paper, a modified extended direct algebraic method with add of symbolic computation is employed to construct bright solitons, dark soliton, periodic solitary wave and elliptic function solutions of two higher order NLSEs such as the resonant NLSE and NLSE with the dual-power law nonlinearity.
Abstract: The propagation of the optical solitons is usually governed by the higher order nonlinear Schrodinger equations (NLSE). In optics, the NLSE modelizes light-wave propagation in an optical fiber. In this article, modified extended direct algebraic method with add of symbolic computation is employed to construct bright soliton, dark soliton, periodic solitary wave and elliptic function solutions of two higher order NLSEs such as the resonant NLSE and NLSE with the dual-power law nonlinearity. Realizing the properties of static and dynamic for these kinds of solutions are very important in various many aspects and have important applications. The obtaining results confirm that the current method is powerful and effectiveness which can be employed to other complex problems that arising in mathematical physics.
TL;DR: In this paper, the authors employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin-Bona-Mahony equation, which has a wide range of applications in modelling long surface gravity waves of small amplitude.
Abstract: In this article, we employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin–Bona–Mahony equation. We obtain some new solutions with the hyperbolic function structures. Benjamin–Bona–Mahony equation has a wide range of applications in modelling long surface gravity waves of small amplitude. We also plot the 2- and 3-dimensional graphics of all analytical solutions obtained in this paper. On the other hand, we analyze the finite difference method and operators, we obtain discretize equation using the finite difference operators. We consider one of the analytical solutions to the Benjamin–Bona–Mahony equation with the new initial condition. We observe that finite difference method is stable when Fourier–Von Neumann technique is used. We also analyze the accuracy of the finite difference method with terms of the errors $$L_{2}$$
and $$L_{\infty }$$
. We use the finite difference method in obtaining the numerical solutions of the Benjamin–Bona–Mahony equation. We compare the numerical results and the exact solution that are obtained in this paper, we support this comparison with the graphic plot. We perform all the computations and graphics plot in this study with the help of Wolfram Mathematica 9.
TL;DR: In this paper, the authors constructed bright and singular optical solitons for the (€ 2+1$$ )-dimensional NLSE and the Hirota equation by utilizing the new sine-Gordon expansion method.
Abstract: This study constructs bright and singular optical solitons the (
$$2+1$$
)-dimensional NLSE and the Hirota equation by utilizing the new sine-Gordon expansion method. The parametric conditions for the existence of valid solitons are given. We give the variations of solitary and periodic wave solutions that depend on the parameters. We present the 2D, 3D and contour graphs to represent the kind of solutions secured in this study.
TL;DR: In this article, the authors reach the dark, bright, mixed dark-bright, singular, mixed singular optical solitons and singular periodic wave solutions to the time-fractional Radhakrishnan-Kundu-Lakshmanan equation.
Abstract: This study reaches the dark, bright, mixed dark-bright, singular, mixed singular optical solitons and singular periodic wave solutions to the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. The parametric conditions that guarantee the existence of valid solitons and other solutions are stated. By choosing some suitable values of parameters, the 2- and 3-dimensional surfaces to some of the reported solutions are plotted. The reported solutions may be useful in expalining the physical meaning of the Radhakrishnan–Kundu–Lakshmanan equation and other related nonlinear models arising in nonlinear sciences.
TL;DR: In this paper, a surface plasmon resonance photonic crystal fiber (PCF) biosensor based on Titanium Nitride (TiN) as a new alternative PLASmonic material is proposed and analyzed.
Abstract: A highly sensitive surface plasmon resonance photonic crystal fiber (PCF) biosensor based on Titanium Nitride (TiN) as a new alternative plasmonic material is proposed and analyzed. The TiN has high stability, high conductivity, and corrosion resistance which make it an ideal material for nanofabrication. The suggested biosensor is analyzed by full vectorial finite element method with perfectly matched layer as boundary conditions. In this paper, the biosensor geometrical parameters are studied to achieve high sensitivity for both polarized modes. A refractive index sensitivity of 7700 and 3600 nm/RIU for quasi-transverse electric and quasi transverse magnetic modes, respectively, are obtained. Additionally, the reported biosensor could be used for detecting an unknown analyte refractive index ranging from 1.32 to 1.34 with a high linearity. Further, the proposed biosensor structure is easy for fabrication using standard PCF fabrication current technologies.
TL;DR: In this article, Feng et al. developed an integral method based on the ring theory of commutative algebra for analytic treatment of fractional differential equations based on conformable fractional derivative.
Abstract: Analytic behavior of fractional differential equations are often seems confusing. Thus, finding comprehensive methods for solving those sounds of high importance. In the present study, Feng’s first integral method which is based on the ring theory of commutative algebra, is developed for analytic treatment fractional differential equations based on conformable fractional derivative. Furthermore, some important nonlinear fractional differential equations, such as Burgers–KdV, Klein–Gordon, KdV–Zakharov–Kuznetsev, and Zakharov–Kuznetsov equations are solved by the proposed approach.
TL;DR: In this paper, a hollow core anti-resonant THz fiber is proposed, which has a simple structure consisting of only ten Topas tubes and achieves high birefringence by introducing two large tubes.
Abstract: A novel high-birefringence hollow-core anti-resonant THz fiber is proposed in this paper. This fiber has a simple structure which consists of only ten Topas tubes. High birefringence is achieved by introducing two large tubes. The first two resonant frequencies are 1.44 and 2.88 THz by fixing tube thickness at 0.09 mm, which makes two low-loss transmission windows exist in the frequency range from 0.8 to 3.0 THz. The lowest loss is 2.10 dB/m occurring at 1.2 THz in the first transmission window and 1.68 dB/m at 2.34 THz in the second transmission window. By optimizing the structure parameters, high birefringence above 7 × 10−4 in the frequency range from 1.0 to 1.24 THz are obtained. The highest birefringence is up to 8.7 × 10−4 at 1.04 THz. Birefringence can be further increased to the order of 10−3 by adjusting the structure parameters at the cost of loss increasing and the bandwidth decreasing. In addition, bent performance of this fiber is also discussed. In addition, this fiber can keep good performance when it is bent for x-direction. At the bend radius of 15 cm, the loss and birefringence has a more slightly change in the first transmission window than the second transmission window. The first transmission window own much better bent-insensitive characteristics.
TL;DR: In this article, the wave solutions of two well-known nonlinear models, namely, the modified Benjamin-Bona-Mahony and the coupled Klein-Gordon equations, are acquired.
Abstract: This study acquires the wave solutions of the two well-known nonlinear models, namely; the modified Benjamin–Bona–Mahony and the coupled Klein–Gordon equations. The modified Benjamin–Bona–Mahony is a nonlinear model that describes the long surface gravity waves of small amplitude and the coupled Klein–Gordon equation describes the quantized version of the relativistic energy–momentum relation. We successfully acquire some new solutions to these models such as kink-type and soliton solutions in complex hyperbolic functions form. We plot the 3D and 2D surface of the all the obtained solutions in this study. The mathematical approach used in this study is the sine-Gordon expansion method.
TL;DR: In this article, the authors obtained dark, bright, combined dark-bright and combined singular soliton solutions to the decoupled nonlinear Schrodinger equation arising in dual-core optical fibers by using the extended sinh-Gordon equation expansion method.
Abstract: The dynamical systems of soliton propagation through optical fibers for trans-continental and trans-oceanic distances is one of the most interesting areas of study. Optical solitons are restrained electromagnetic waves that stretch in nonlinear dispersive media and allow the intensity to remain unchanged due to the balance between dispersion and nonlinearity effects. In this study, we successfully acquire dark, bright, combined dark–bright, singular and combined singular soliton solutions to the decoupled nonlinear Schrodinger equation arising in dual-core optical fibers by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid soliton solutions are given. We discuss how change in parameters affect the solitons transmission. We present the 2D, 3D and the contour graphs to some of the reported solutions.