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With an increasing demand for larger bandwidth and solving the problem of internet bottleneck, the formation of solitons with fractional temporal evolution has prompted substantial attention in the field of optics. In this paper, we have used three methods namely, the new mapping method, the unified Riccati equation method, and the new modified sub-ODE method to construct the new soliton solutions to the perturbed nonlinear Schrödinger equation (NLSE) with fractional temporal evolution. Two types of nonlinearities are considered namely, the Kerr law and the power law for all the three utilized methods. Existence criteria for all soliton solutions are presented. We provide two- and three-dimensional graphical illustrations of some acquired solutions. This work also proves the reliability, power, and effectiveness of these new analytical techniques for solving other nonlinear fractional differential equations.

In this work, we concern with the soliton solutions to an optical twin-core coupler that occur with four sorts of nonlinearities viz. Kerr law, power law, parabolic law, and dual power-law nonlinearity. The twin-core couplers with the nearest neighbor linear coupling are considered and bright, dark, and singular optical solitons are extracted with the aid of the integration algorithm of the new mapping method. The appropriate constraint conditions are naturally emerged to secure the existence of these solitons. For physical classification of the newly acquired solutions, we have expressed them graphically to describe the physical interpretation of distinct shaped solutions more evidently.

This work theoretically analyzed the role played by intrinsic degrees of freedom of the colliding partners and/or nucleon transfer channel in the fusion dynamics of [Formula: see text]C + [Formula: see text]Zr and [Formula: see text]C + [Formula: see text]Pt reactions. The theoretical calculations for the chosen reactions have been done by using symmetric–asymmetric Gaussian barrier distribution (SAGBD) model, coupled channel code CCFULL and one-dimensional Wong formula. The fusion excitation functions for these reactions are enhanced at energies lying in the sub-barrier regions with respect to calculations based on one-dimensional Wong formula. Such enhancement can be ascribed to the involvement of the low-lying inelastic surface excitations of the colliding systems and some other favored channels. The CCFULL calculations consider the couplings to low lying vibrational states of target isotopes and hence intimate the dominance of the collective excitations associated with the fusing partners. For all the studied reactions, the SAGBD model adequately addresses the sub-barrier fusion dynamics and thus the impacts of nuclear structure degrees of freedom associated with the collision partners are intrinsically incorporated via single Gaussian type of weight function. In the SAGBD approach, the nuclear structure effects are analyzed in terms of channel coupling parameter [Formula: see text] and percentage reduction of effective fusion barrier [Formula: see text] with reference to the Coulomb barrier. The larger values for [Formula: see text] and [Formula: see text] imply that the channel couplings due to internal structure of participants are significant at below barrier energies. The CCFULL calculations and SAGBD predictions consistently explain the fusion dynamics of chosen systems and hence reveal that both models have almost similar predictive power with respect to reported fusion data.

The role of the nuclear structure degrees of freedom associated with the colliding nuclei and the energy dependence in the nucleus–nucleus interaction potential in fusion dynamics of various projectiles (16O, 27Al and 37Cl) with the Ge-isotopes (74,76Ge) has been investigated. Fusion cross-sections obtained through energy dependent interaction potential (EDWSP model) and coupled channel theory show quite similar behavior in near and above barrier energy regions. In coupled channel view, the couplings to inelastic surface excitations associated with the fusing nuclei results in large fusion enhancement at sub-barrier energies. The consideration of spherical shape for targets (74,76Ge-isotopes) along with their relevant internal structure degrees of freedom in coupled channel approach is sufficient to get a good description to observed fusion enhancement of the studied reactions. In contrast, the energy dependent interaction potential modifies the barrier profile of interaction barrier and subsequently reduces the effective fusion barrier between the colliding systems, henceforth, predicts substantially large fusion enhancement at below barrier energies in comparison to the expectations of the one dimensional barrier penetration model. Furthermore, χ 2 -analysis and the ratio plots σ T h σ E x p for outcomes of the EDWSP model suggested that the model calculations provide a consistent fit to fusion data within 10%. For the chosen reactions, only at 6 fusion data points out of 75 fusion data points does the deviation exceed 5% while 69 fusion data points lie within 5% and hence, the EDWSP model reasonably explained the above barrier fusion data within 5% with a probability more than 90%.

This paper uses the new modified sub-ODE method, the new extended auxiliary equation method, and the new Jacobi elliptic function expansion method to revisit the (2+1)-dimensional coupled nonlinear Schr¨odinger equation with cascading effect. As a consequence, dark, bright, kinkantikink, singular solitons, Weierstrass elliptic function, doubly periodic, and complex optical soliton solutions are retrieved. All solutions are described, along with the existence criterion on the parameters. As solitons are used for data transfer, the obtained results may be found usage in optical couplers, birefringed fibres, and optoelectronic devices. A comparison of the obtained results with those found in the literature is given. The dynamical behaviour of some of the obtained solutions has been explored for suitable choices of the parameters. Using the property of Hamiltonian systems, the solitons stability is determined.

In this paper, we have investigated the sensitivity of fusion cross section and the barrier distribution data with respect to the choice of the potential parameters. To ascertain this, the fusion dynamics of $^{16}\mathrm{O}+^{70,72,73,74,76}\mathrm{Ge}$ and $^{18}\mathrm{O}+^{74}\mathrm{Ge}$ reactions at near and sub-barrier energies have been examined. In the present study, we have used the one-dimensional Wong formula, symmetric-asymmetric Gaussian barrier distribution (SAGBD) model, and coupled channel approach. For theoretical estimations, the Woods-Saxon form of nuclear potential has been adopted, and it has been found that both the fusion cross section and barrier distribution are quite sensitive to the choice of potential parameters in the entire range of incident energies. Calculations based on the Wong formula are significantly smaller than the sub-barrier fusion data, which has clearly suggested that the couplings to the nuclear structure's degrees of freedom of the colliding nuclei are needed to reproduce the experimental data. On the other hand, in the SAGBD model calculations the multidimensional character of nucleus-nucleus potential has been incorporated and it has reasonably recovered both the fusion cross-section data and the barrier distribution data of $^{16}\mathrm{O}+^{70,72,73,74,76}\mathrm{Ge}$ and $^{18}\mathrm{O}+^{74}\mathrm{Ge}$ reactions. In the SAGBD model, channel coupling parameter $\ensuremath{\lambda}$ has been determined from barrier distribution and hence $\ensuremath{\lambda}$ is related to the various channel coupling effects which have arisen because of the nuclear structure of the participating nuclei. The increase in $\ensuremath{\lambda}$ with an increase in target isotopic mass has clearly indicated that there are shape transition effects in Ge isotopes, which change their shape from spherical symmetry to prolate deformed shape as one moves from $^{70}\mathrm{Ge}$ to $^{76}\mathrm{Ge}$. Similarly, the percentage reduction of effective fusion barrier ${V}_{\mathit{CBRED}}$ with respect to uncoupled Coulomb barrier has also shown an increasing trend with the increase in target isotopic mass. In order to confirm the shape transition effects, the coupled channel calculations have been performed by using the code CCFULL. The present coupled channel analysis has clearly pointed out the importance of nuclear structure degrees of freedom of participating nuclei in the sub-barrier region. Similar results have also been evident from SAGBD analysis. Hence the model predictions incorporate the contributions from vibrational excitations and/or static deformation in the fusion of $^{16}\mathrm{O}+^{70,72,73,74,76}\mathrm{Ge}$ reactions. For the $^{18}\mathrm{O}+^{74}\mathrm{Ge}$ reaction, the vibrational excitations along with the neutron transfer channel have led to larger sub-barrier fusion enhancement and their influences have been automatically included in the SAGBD calculations. Furthermore, $^{16}\mathrm{O}+^{76}\mathrm{Ge}$ and $^{18}\mathrm{O}+^{74}\mathrm{Ge}$ reactions have formed the same compound nucleus $^{92}\mathrm{Zr}$ via different entrance channels. Their comparative study has indicated that the effect of the neutron transfer channel has dominated over the entrance channel mass asymmetry effects. In addition, the minimum value of ${\ensuremath{\chi}}^{2}$ that has been obtained for the SAGBD calculations is consistent with that of the coupled channel calculations.

In this work, Nuclear Energy Density Functional Theory (NEDFT) has been used to investigate the various structural properties of stable and weakly bound and/or Halo structure nuclei. The different Skyrme interactions like SKM*, SKP Sly4 are utilized to study the structural properties of 4He, 6He and 8He nuclei. Further all the density functionals predict approximately similar behaviors for these nuclei and the calculations based on adopted Skyrme interactions indicate that 6He and 8He nuclei exhibits neutron halo structure.

This paper examined the fusion of 28Si with 90,96Zr-target by opting the energy dependent Woods-Saxon potential (EDWSP) model and code CCFULL. The CCFULL analysis suggested that for 28Si + 90Zr system, the vibrational excitations of 90Zr-isotope have been found to be very important while for 28Si + 96Zr system, in addition to multiphoton vibrational states of target, the considerations of nucleon transfer are essentially needed for the adequate addressal of the observed fusion data. In contrast, predictions based on EDWSP model appropriately recovered the fusion data of 28Si + 90,96Zr systems at sub-barrier realm. The EDWSP based results do not include various channel coupling effects explicitly but being the energy dependent nature of Woods-Saxon potential (WSP), the calculational results intrinsically incorporate the nuclear effects of the fusing systems. This unambiguously clarified that the energy dependent nature of the EDWSP generates analogous barrier modulations as emerged from the CCFULL analysis.

The fusion-fission dynamics of 12C + 232Th reaction at energies in sub-barrier realm are analyzed by using one-dimensional Wong formula and symmetric-asymmetric Gaussian barrier distribution (SAGBD) model. For this, we examined the fusion cross-sections data and related barrier distribution. Calculations based on simple Wong formula fails to explain the fusion data specially at below barrier energies, while SAGBD model predictions qualitatively as well as quantitively reproduced the fusion dynamics of studied reaction. The effects of coupling in SAGBD model are calculated in terms of channel coupling parameter λ, which point towards the reduction of height of the effective fusion barrier and consequently recovered large enhancement in fusion cross-sections in comparison with estimations obtained by using the Wong formula. Furthermore, the percentage reduction of the nominal fusion barrier due to involvement of nuclear structure effects, is described in terms of VCBRED. For the studied system, χ2-analysis shows small value for the SAGBD based results and hence clearly indicates the consistency of the present approach.

By utilizing the three distinctive approaches specifically, the extended Fan sub-equation method, the exp[ $$-G(\xi )$$ ]-function expansion method, and the fractional transformation method, the traveling waves and soliton solutions of the (4+1)-dimensional nonlinear Fokas equation are extracted. Meanwhile, some parametric constraint conditions are described. The acquired solutions are singular and nonsingular soliton solutions, periodic solutions, breather solution, rational solutions, trigonometric periodic wave solutions, hyperbolic solutions, Weierstrass, and Jacobi elliptic doubly periodic wave solutions. The dynamics of some of the obtained solutions are investigated and described in 2-dimensional figures by choosing appropriate parameter values. The comparison of our obtained results with the other solutions in literature shows that the obtained solutions of this paper are new and have not been formulated before by other techniques. We believe that all these results are useful to enrich the knowledge of the important physical phenomenon characterized by the Fokas equation. The reported solutions illustrate the straightforwardness, reliability, and effectiveness of the used techniques that can be further employed to higher-dimensional nonlinear evolution equations.

This work theoretically explores the role of neutron transfer channels and/or collective inelastic surface excitations in the fusion of $^{28}\mathrm{Si}$ with $^{90,92,94,96}\mathrm{Zr}$ targets by using the coupled channel theory and the energy dependent Woods-Saxon potential (EDWSP) model. The series of $^{90,92,94,96}\mathrm{Zr}$ targets is quite interesting due to the fact that the possibilities of neutron transfer channels with positive ground state $Q$ values increase with the increase of isotopic mass. For $^{28}\mathrm{Si}+^{90}\mathrm{Zr}$ reaction, the influences of inelastic surface excitations turned out to be important and coupling of such channels to their relative motion reproduced the experimental data. In the case of $^{28}\mathrm{Si}+^{92}\mathrm{Zr}$ reaction, in addition to consideration of low lying states such as ${2}^{+}$ and ${3}^{\ensuremath{-}}$ vibrational states of colliding nuclei, the coupling to two neutron pickup channels is necessarily required to address the sub-barrier fusion anomalies. For $^{28}\mathrm{Si}+^{94,96}\mathrm{Zr}$ reactions, the inclusions of multiphonon states of type ${2}^{+}$ and ${3}^{\ensuremath{-}}$ of colliding nuclei were not able to reproduce the fusion enhancement particularly at below barrier energies. In this case, neutron transfer channels with positive ground state $Q$ value play a crucial role in the enhancement of fusion cross-section data at sub-barrier energies and therefore must be included in the coupled channel description. In distinction, in the EDWSP model, the energy dependence in the nucleus-nucleus potential causes barrier modification effects and subsequently induces a barrier lowering phenomenon. In this way, the EDWSP based outcomes reasonably address the sub-barrier fusion anomalies of $^{28}\mathrm{Si}+^{90,92,94,96}\mathrm{Zr}$ reactions and thus impacts of dominant intrinsic channels are intrinsically included due to the dynamical nature of the energy dependent interaction potential. For studied systems, the EDWSP outcomes are able to achieve an agreement with the portion of above barrier fusion data within 10% with a probability greater than 90%. Within this model, 33 fusion data points out of 38 fusion data points lie within 5%. Only five fusion data points lie within 10% and thereby the EDWSP model adequately addresses the fusion anomalies of the chosen reactions. The smaller values of ${\ensuremath{\chi}}^{2}$ analysis for the EDWSP calculations, which range from 2.82 to 3.14, indicate that the model predictions appropriately describe the observed fusion dynamics of the studied reactions.

We have investigated the fusion dynamics of 12C + 92Zr, 16O + 92Zr, 28Si + 92Zr, 35Cl + 92Zr, 16O + 60Ni, 18O + 58Ni, 12C + 194,198Pt, 16O + 144,148,154Sm and 17O + 144Sm reactions in the close vicinity of the Coulomb barrier. For this purpose, we have opted one-dimensional Wong formula, coupled channel approach, and symmetric-asymmetric Gaussian barrier distribution (SAGBD) model to analyze the fusion data for above mentioned reactions. For all studied systems, the theoretical results obtained from the one-dimensional Wong formula are unable to reproduce the fusion data especially at incident energies lying below the nominal barrier. This appeals for the inclusion of intrinsic degrees of freedom which have been originated from the nuclear structure of fusing nuclei, and thus essentially required to explain the experimental data at energies lying in the sub-barrier realm. For the chosen reactions, the coupled channel analysis suggests that the coupling of relative motion with the inelastic surface excitations and/or nucleon transfer channels associated with the reaction partners have produced a remarkable fusion enhancement at energies lying near and below the nominal barrier. It is particularly intriguing that the SAGBD model can replicate the impacts of dominant intrinsic reaction channels such as multi-phonon quantum states and/or nucleon transfer channels via Gaussian type of weight function. The predictions of the SAGBD approach appropriately reproduce the fusion cross-sections data and related barrier distributions data for the selected reactions due the fact that the multi-dimensional nature induces the barrier lowering effects and subsequently reduces the effective fusion barrier between the participants. In the SAGBD approach, the channel coupling effects are quantitively measured in terms of channel coupling parameter ( λ ) and percentage decrement in height of effective fusion barrier with respect to the Coulomb barrier ( V C B R E D ) . Furthermore, the recovered χ 2 -values for selected reactions are found to be smaller and hence clearly demonstrate the applicability of the model outcomes to explore the fusion dynamics of the studied systems.

The role of barrier modifications and the relevant nuclear structure effects in the fusion of the $$ {}_8{}^{16}O+{}_{62}{}^{144,148,150,152,154}Sm $$ and $$ {}_3{}^{6,7}Li+{}_{62}{}^{152}Sm $$ systems is analyzed within the context of the energy-dependent Woods-Saxon potential model (EDWSP model) and the coupled channel model. For the $$ {}_8{}^{16}O+{}_{62}{}^{144,148,150,152,154}Sm $$ reactions, where the colliding pairs are stable against breakup, the collective excitations and/or static deformations are sufficient to account for the observed fusion enhancement. In contrast, the model calculations overpredict the complete fusion data at above - barrier energies for the $$ {}_3{}^{6,7}Li+{}_{62}{}^{152}Sm $$ systems, where the importance of projectile breakup effects has been pointed out. Due to the low threshold of the alpha-breakup channel, the weakly bound projectiles $$ \left({}_3{}^{6,7}Li\right) $$ break up into charged fragments before reaching the fusion barrier and consequently the complete fusion cross section is suppressed by 28% (25%) in the $$ {}_3{}^6Li+{}_{62}{}^{152}Sm\;\left({}_3{}^7Li+{}_{62}{}^{152}Sm\right) $$ reaction with respect to predictions of coupled channel calculations. However, the EDWSP model based calculations can minimize the suppression factor by as much as of 13% (8%) in the $$ {}_3{}^6Li+{}_{62}{}^{152}Sm\;\left({}_3{}^7Li+{}_{62}{}^{152}Sm\right) $$ reaction with reference to the predictions made by the coupled channel calculations. Therefore, the complete fusion data of the $$ {}_3{}^6Li+{}_{62}{}^{152}Sm\;\left({}_3{}^7Li+{}_{62}{}^{152}Sm\right) $$ reaction at above - barrier energies is reduced by 15% (17%) with respect to the expectations of the EDWSP model. The extracted suppression factors for the studied reactions are due to the modifications of the barrier profile as a consequence of the energy - dependence in nucleus-nucleus potential, and thus greater barrier modifications occur for more weakly bound system, which in turn, confirms the breakup of projectile in the incoming channel.

We explore a family of exact traveling and localized soliton solutions of the nonlinear integrable (4+1)-dimensional Fokas equation by different approaches, namely, the Jacobian-function method, the Pades type transformation, He’s semi-inverse variational technique and sine–cosine or triangle function approach. Some new exact traveling wave solutions of physical interest involving some constraint conditions are reported. The reported solutions are Jacobi doubly periodic wave solutions, fractional soliton, localized soliton when parameters are taken to be special values of doubly periodic functions and complex Bloch solitons. These obtained solutions may facilitate us in understanding the dynamical behavior of physical phenomenon governed by nonlinear integrable Fokas equation and show the applicability and efficacy of the used approaches that can be applied for higher dimensional nonlinear integrable equations as well as linear ones in mathematical physics. The differences and similarities between the used distinct approaches are discussed. To exhibits, the dynamical behavior of reported solutions, the two and three-dimensional structures are numerically simulated.

In this study, we intend to analyze the traveling and several other solitary wave solutions in the nonlinear low-pass electrical transmission line model using the new mapping method, the new extended auxiliary equation method, and the extended Kudryashov method. A type of traveling and solitary wave solutions emerge, consisting of hyperbolic function, trigonometric, rational, periodic, and doubly periodic solutions that reflect kink, anti-kink wave solitons, bright-dark optical solitons, singular solitons, and other traveling waves. The three integration techniques applied are efficient, effective, and versatile for the creation of new bright, dark, singular, and non-singular periodic and solitary wave propagation solutions in nonlinear low-pass electrical transmission lines. To see the extant physical significance of the considered equation, we present some 2D and 3D figures for some solutions. We compare the obtained results with those obtained in the literature. We investigate and demonstrate the stability of the soliton solutions.

This work explored the fusion dynamics of 3 6 L i + 27 59 C o , 4 9 B e + 50 124 S n , 4 9 B e + 82 208 P b , 20 40 C a + 50 124 S n and 6 12 C + 82 208 P b reactions by using energy dependent Woods–Saxon potential model (EDWSP model) and coupled channel model. The roles of projectile breakup channel on fusion process are directly reflected from fusion mechanism of 3 6 L i + 27 59 C o , 4 9 B e + 50 124 S n and 4 9 B e + 82 208 P b reactions. Whereas, the impacts of collective excitations and/or nucleon transfer channels on fusion process are evident from analysis of 20 40 C a + 50 124 S n and 6 12 C + 82 208 P b reactions. The total fusion data (sum of complete and incomplete fusion data) of 3 6 L i + 27 59 C o reaction is reasonably recovered by the EDWSP model calculations which in turn indicate that total fusion excitation function data is not suppressed with reference to theoretical predictions. In contrast, for 4 9 B e + 50 124 S n ( 4 9 B e + 82 208 P b ) reaction, the above barrier fusion data is suppressed with respect to the expectations of coupled channel approach and single barrier penetration model by 28% (32%). However, the EDWSP model based calculations reduce the magnitude of suppression factor by 8% (12%) for 4 9 B e + 50 124 S n ( 4 9 B e + 82 208 P b ) reaction and consequently the fusion data at above barrier energies is inhibited for both reactions by 20% with reference to the EDWSP model calculations. This fusion suppression can be correlated with loosely bound nature of the projectile. On the other hand, the above barrier fusion data of 20 40 C a + 50 124 S n and 6 12 C + 82 208 P b reactions is not inhibited with respect to the predictions of coupled channel approach and EDWSP method, henceforth, confirms stability of projectiles against breakup effects.

This paper analyzed the role of intrinsic degrees of freedom of colliding nuclei in the enhancement of sub-barrier fusion cross-section data of various heavy ion fusion reactions. The influences of inelastic surface vibrations of colliding pairs are found to be dominant and their couplings result in the significantly larger fusion enhancement over the predictions of the one dimensional barrier penetration model at sub-barrier energies. The theoretical calculations are performed by using energy dependent Woods–Saxon potential model (EDWSP model) in conjunction with the one dimensional Wong formula. The effects of dominant intrinsic channels are entertained within framework of the coupled channel calculations obtained by using the code CCFULL. It is quite interesting to note that the energy dependence in Woods–Saxon potential simulates the effects of inelastic surface vibrational states of reactants wherein significantly larger value of diffuseness parameter ranging from a = 0.85fm to a = 0.95fm is required to address the observed fusion excitation function data of the various heavy ion fusion reactions.

The impacts of breakup channel on fusion of L 6 i with Z 90 , 96 r -nuclei are investigated within the context of the coupled channel model and the energy dependent Woods-Saxon potential (EDWSP) model. For studied reactions, the complete fusion (CF) cross-sections data at above barrier energies are found to be suppressed with respect to the calculated results. In case of L 6 i + 90 Z r ( L 6 i + 96 Z r ) reaction, the CF data at above barrier energies is found to be less by 34% (25%) with reference to the estimations of the coupled channel approach. However, EDWSP model calculations predict fusion suppression up to 20% (10%) for L 6 i + 90 Z r ( L 6 i + 96 Z r ) reaction. The extracted suppression factor for L 6 i + 90 Z r ( L 6 i + 96 Z r ) reaction is smaller by 14% (15%) than the reported value. Such effects can be accredited to the low threshold of α-breakup channel associated with the weakly bound system and hence may be responsible for incomplete fusion (ICF) products. Therefore, the total fusion (TF) cross-sections that are the sum of complete fusion (CF) and incomplete fusion (ICF) yields, are not expected to reduce in the above barrier regions when compared with no-breakup case. In this regard, the TF cross-sections data are analyzed within the view of the EDWSP model and so obtained calculations have close agreement with the TF cross-sections data. The EDWSP based results appreciably explained CF, ICF and TF cross-sections data over a wide range of energies around the Coulomb barrier. Although, the cluster-transfer or stripping process from the ground state of weakly bound nucleus L 6 i may significantly contribute to ICF yields, therefore, the presence of ICF product is an indication of the partial absorption of projectile by target, henceforth, reduces the CF cross-section data in above barrier regimes.

This work emphasized the role of the projectile breakup channel by studying the complete fusion (CF) and incomplete fusion (ICF) dynamics of [Formula: see text] reactions. The theoretical calculations for the chosen reactions have been done by opting for the coupled channel approach and the energy dependent Woods–Saxon potential (EDWSP) model. The below barrier fusion enhancements of the studied reactions are reasonably addressed by the outcomes of the adopted models, which in turn can be attributed to the couplings of nuclear structure degrees of freedom of the collision partners to their relative motion. In contrast, at above barrier energies, the CF cross-section data of the chosen reactions are found to be suppressed significantly when compared with the predictions made by using the present models. Interestingly, the fusion suppression factors of the given reactions can be minimized considerably with respect to the reported value when it is analyzed within the framework of the EDWSP model. For instance, in case of [Formula: see text] ([Formula: see text] reaction, the magnitude of fusion suppression factor is minimized up to 7% (13%) relative to the reported value whereas for [Formula: see text] ([Formula: see text] reaction, the fusion suppression factor is found to be less by 7% (8%) with reference to the reported value. Such suppression effects can be correlated with the low breakup threshold of alpha breakup channel associated with the loosely bound projectile. The projectiles being weakly bound systems split into two charged fragments and either of the breakup components is absorbed by the target resulting in the reduction of incoming flux going into fusion channel. The flux lost from the CF channel appears in the form of ICF yields. For [Formula: see text], total fusion (TF) cross-sections that are sum of CF and ICF cross-sections are also analyzed in conjunction with the EDWSP model and thus reasonably explained by the model calculations. In order to identify the ICF contribution, the ratio of ICF/TF cross-section data of [Formula: see text] reaction has been examined and thus properly addressed by using the EDWSP model. The presence of ICF component in TF cross-section clearly pointed out the breakup of projectile due to its loosely bound nature prior to the Coulomb barrier. Although ICF data of other systems are not available in the literature, a similar behavior is expected for ICF and TF data for [Formula: see text] and [Formula: see text] reactions.

Limitations of the static Woods–Saxon potential and the applicability of the energy-dependent Woods–Saxon potential (EDWSP) model within the framework of one-dimensional Wong formula to explore the sub-barrier fusion data are highlighted. The inelastic surface excitations of the fusing nuclei are found to be dominating in the enhancement of sub-barrier fusion excitation function data and the effects of such dominant vibrational states are exploited through the coupled channel calculations obtained by using the code CCFULL. It is worth mentioning here that the influence of multiphonon vibrational states of the reactants can be simulated by introducing the energy dependence in the nucleus–nucleus potential.

The fusion dynamics of 16 S+40 Zr reaction at near and sub-barrier energies is investigated within the context of different theoretical approaches. The various theoretical models like one-dimensional Wong formula, l-summed ex- tended Wong formula, the energy-dependent Woods-Saxon potential model (EDWSP model), and coupled channel for- mulation have been used to address the impacts of nuclear structure degrees of freedom of the colliding pairs. The roles of different Skyrme forces along with Wong formalism are also tested in the analysis of the sub-barrier fusion dynamics of the 16 S+40 Zr reaction. The influence of the low-lying sur- face vibrational states of the collision partners is investigated within the framework of coupled channel calculations per- formed by the code CCFULL. In the present work, it has been observed that the EDWSP model introduces barrier modifica- tion effects somewhat similar to those of the coupled channel approach, as well as those of using different Skyrme forces and hence it reasonably addresses the observed fusion data of 16 S+40 Zr reaction in the close vicinity of the Coulomb barrier.

This work theoretically investigates the role of the projectile degrees of freedom on the fusion dynamics of various heavy-ion fusion reactions. The impact of the projectile breakup channel is studied for the fusion mechanism of the 4 9 Be + 39 89 Y, 6 12 C + 39 89 Y, and 16 32,34 S + 39 89 Y reactions within the view of the coupled channel approach and the energy-dependent Woods-Saxon potential model (EDWSP model). The above-barrier fusion cross-section data of the 4 9 Be + 39 89 Y reaction is suppressed by about 20 % with respect to the theoretical predictions of the coupled channel approach and the single barrier penetration model while this suppression factor is reduced to 10 % within the context of the EDWSP model calculations. Such fusion hindrance at above-barrier energies can be understood in terms of the projectile breakup effects that arise due to its low breakup threshold. However, the observed fusion enhancement of the 6 12 C + 39 89 Y and 16 32,34 S + 39 89 Y reactions, wherein the colliding pairs are stable against breakup, is adequately explained by the EDWSP model and the coupled channel approach in the whole range of energy around the Coulomb barrier. This reveals that the energy dependence in the nucleus-nucleus potential governs barrier modification effects (barrier height, barrier position, barrier curvature) in closely similar way as reflected from the coupled channel formulation.

This article analyzes the validity of static Woods–Saxon potential and the energy-dependent Woods–Saxon potential (EDWSP) to explore the specific features of fusion dynamics of [Formula: see text] and [Formula: see text] systems. The intrinsic degrees of freedom, such as inelastic surface excitations, play a crucial role in the enhancement of sub-barrier fusion excitation functions over the expectations of the one-dimensional barrier penetration model. Role of dominant intrinsic degrees of freedom of collision partners are entertained within the context of coupled channel calculations. Furthermore, the one-dimensional Wong formula using static Woods–Saxon potential fails miserably to describe the fusion enhancement of [Formula: see text] and [Formula: see text] systems. However, the Wong formula along with the EDWSP model accurately explains the observed fusion enhancement of [Formula: see text] reactions. In the fusion of [Formula: see text] reaction, the above-barrier fusion data are suppressed by a factor of 0.66 with reference to the EDWSP model calculations while the below-barrier fusion data are adequately addressed by the EDWSP model and the coupled channel calculations. Therefore, the coupled channel calculations and the EDWSP model calculations reasonably describe the observed fusion mechanism of [Formula: see text] and [Formula: see text] reactions. This suggests that the energy dependence in the Woods–Saxon potential model introduces similar kinds of barrier modification effects (barrier height, barrier position, and barrier curvature) as reflected from the coupled channel calculations. In the EDWSP model calculations, significantly larger values of diffuseness ranging from a = 0.86 to 0.94 fm, which is much larger than a value extracted from the elastic scattering analysis, are needed to address the sub-barrier fusion data.

We have analyzed the role of barrier modification effects (barrier height, barrier position, barrier curvature) introduced due to the energy-dependent Woods–Saxon potential model (EDWSP model) and the coupled channel model on the sub-barrier fusion dynamics of $$ {}_{16}^{32,36} {\text{S}} + {}_{40}^{90,96} {\text{Zr}} $$ reactions. The influence of inelastic surface excitations of colliding pairs and multi-neutron transfer channels is found to be a dominant mode of couplings. The coupling of relative motion of colliding nuclei to these dominant intrinsic degrees of freedom leads to a substantially large fusion enhancement at below-barrier energies over the expectations of one-dimensional barrier penetration model. The coupled channel calculations based upon static Woods–Saxon potential must include the internal nuclear structure degrees of freedom of colliding nuclei for complete description of experimental data. On the other hand, theoretical calculations based upon the EDWSP model along with Wong formula provide a complete description of sub-barrier fusion enhancement of various heavy-ion fusion reactions. In EDWSP model calculations, significantly larger values of diffuseness parameter ranging from a = 0.98 fm to a = 0.85 fm are required to address the observed sub-barrier fusion enhancement of $$ {}_{16}^{32,36} {\text{S}} + {}_{40}^{90,96} {\text{Zr}} $$ reactions. Furthermore, within the context of EDWSP model, it is possible to achieve an agreement with the experimental fusion cross-sectional data within 10 %. For four heavy-ion fusion reactions, only at 4 fusion data points out of 90 fusion data points deviates exceeding 5 %, while 86 fusion data points lie within 5 % and hence the EDWSP model is able to account the above-barrier portion of the fusion cross-sectional data within 5 % with a probability greater than 90 %.

In the present work, the fusion of symmetric and asymmetric projectile–target combinations are deeply analyzed within the framework of energy dependent Woods–Saxon potential model (EDWSP model) in conjunction with one dimensional Wong formula and the coupled channel code CCFULL. The neutron transfer channels and the inelastic surface excitations of collision partners are dominating mode of couplings and the coupling of relative motion of colliding nuclei to such relevant internal degrees of freedom produces a significant fusion enhancement at sub-barrier energies. It is quite interesting that the effects of dominant intrinsic degrees of freedom such as multi-phonon vibrational states, neutron transfer channels and proton transfer channels can be simulated by introducing the energy dependence in the nucleus–nucleus potential (EDWSP model). In the EDWSP model calculations, a wide range of diffuseness parameter ranging from a = 0.85 fm to a = 0.97 fm , which is much larger than a value ( a = 0.65 fm ) extracted from the elastic scattering data, is needed to reproduce sub-barrier fusion data. However, such diffuseness anomaly, which might be an artifact of some dynamical effects, has been resolved by trajectory fluctuation dissipation (TFD) model wherein the resulting nucleus–nucleus potential possesses normal diffuseness parameter.

The dynamics of complete and incomplete fusion of 6,7Li, 15N and 16O with a common target (209Bi) around the Coulomb barrier are analyzed within the context of the coupled channel formulation and the energy dependent Woods-Saxon potential (EDWSP) model. The calculated results are compared with experimental fusion cross-sections and it has been shown that complete fusion (CF) data of weakly bound projectile with a heavy target (209Bi) gets suppressed at above barrier energies. In the case of the 6Li + 209Bi (7Li + 209Bi) reaction, the CF data at above barrier energies is reduced by 34% (26%) with reference to the expectations of the coupled channel approach. However, the theoretical estimations due to the EDWSP model can minimize the suppression factor by 9% with respect to the reported value and consequently the portion of above barrier CF cross-section data of 6Li + 209Bi (7Li + 209Bi) reaction is suppressed by 25% (17%) when compared with the present model calculations. This fusion inhibition can be correlated with the low breakup threshold of projectile which in turn breaks up into two fragments in the entrance channel prior to fusion barrier. The total fusion (TF) data, which is sum of complete fusion (CF) data and incomplete fusion (ICF) data, is not suppressed when compared with the predictions of the theoretical approaches and thus breakup channel has very little influence on the total fusion cross-sections. Although the breakup fragments appeared in both reactions, the enhanced suppression effects observed for the lighter projectile can be correlated with its low binding energy associated with the $\alpha$ -breakup channel. Further the outcomes of the EDWSP model reasonably explained the ICF contribution appeared in the fusion of 6,7Li + 209Bi reactions. In contrast to this, the observed fusion dynamics of 15N + 209Bi and 16O + 209Bi reactions, wherein the collective excitations such as two phonon, three phonon vibrational states contribute to produce below barrier fusion enhancement, has been adequately explored by the adopted models, and henceforth ensures the stability of well bound nuclei against breakup effects.

The complete fusion (CF) and incomplete fusion (ICF) cross-sections are estimated for 6, 7Li + 159Tb reactions using the energy-dependent Woods-Saxon potential model (EDWSP model) and dynamical cluster-decay model (DCM). The CF data of the 6Li + 159Tb(7Li + 159Tb) reaction at above barrier energies is suppressed with reference to expectations of the EDWSP model by 25% (20%) which is smaller than the reported data by ∼ 9% (6%). This suppression is correlated with the projectile breakup effect. The projectiles 6,7Li are loosely bound systems, which may break up into charged fragments prior to reaching the fusion barrier and subsequently one of the fragment is captured by the target leading to the suppression of fusion data at above barrier energies. The sum of CF and ICF, which is termed as total fusion cross-section (TF), removes the discrepancies between theoretical predictions and the above barrier complete fusion data and hence is adequately explained via the EDWSP model over a wide range of energy spread across the Coulomb barrier. In addition to fusion, the decay mechanism of 6Li + 159Tb reaction is studied within the framework of the dynamical cluster-decay model (DCM). The breakup of the projectile (6Li) in the entrance channel indicates the presence of ICF, which is investigated further using the collective clusterization approach of DCM. The present theoretical analysis suggests that a larger barrier modification is needed to address the fusion data of chosen reactions in the below barrier energy region.

This work examined the fusion dynamics of [Formula: see text] and [Formula: see text] reactions within the framework of the static Woods–Saxon potential model, the energy dependent Woods–Saxon potential (EDWSP) model and coupled channel formulation. The effects of inelastic surface excitations, static deformation of colliding pairs and /or neutron transfer channels on fusion process are investigated through the coupled channel method. The calculations based upon static Woods–Saxon potential in conjunction with one-dimensional Wong formula strongly under predict the fusion data of [Formula: see text] and [Formula: see text] reactions at sub-barrier energies. However, such discrepancies are removed if one uses couplings to nuclear structure degrees of freedom of reacting nuclei. The coupled channel calculations obtained by considering the vibrational nature of the colliding nuclei fairly reproduce the fusion data of [Formula: see text] reactions. For this reaction, the neutron transfer channels, which are expected to influence strongly the fusion yields at below barrier energies, in reality contribute very weakly to fusion process. While in case of [Formula: see text] reaction, the consideration of vibrational couplings as well as the rotational couplings for target provides a reasonable explanation to the fusion cross-section data at near and above barrier energies. In distinction, the energy dependence in the nucleus–nucleus potential causes barrier modulation effects and subsequently modifies the barrier profile of the interaction barrier in such a way that the effective fusion barrier between the colliding pair reduces. This ultimately brings larger fusion cross-sections over the outcomes of one-dimensional barrier penetration model and the EDWSP model based calculations appreciably explained the fusion dynamics of chosen reaction at energy spanning around the Coulomb barrier. Both models (EDWSP and coupled channel model) lead to barrier lowering effects and modeled quantum tunneling in different way, henceforth, adequately explore the fusion dynamics of the studied reactions in near and above barrier energy regions.

We have analyzed the fusion dynamics of $_{20}^{40}\mathrm{Ca}+_{28}^{58,64}\mathrm{Ni}$ reactions by using the energy dependent Woods-Saxon potential (EDWSP) model and coupled channel model and subsequently the decay patterns of $_{48}^{98,104}\mathrm{Cd}^{*}$ nuclei are governed via the dynamical cluster-decay model (DCM). The influence of intrinsic degrees of freedom of colliding pairs, such as low lying surface vibrations and neutron transfer channels, are entertained within the context of coupled channel calculations. Interestingly, the energy dependence in the Woods-Saxon potential induces barrier modification effects (barrier height, barrier position, barrier curvature) in a somewhat similar way as that for coupled channel approach, hence adequately explains the observed fusion dynamics of $_{20}^{40}\mathrm{Ca}+_{28}^{58,64}\mathrm{Ni}$ reactions. In addition, the decay analysis of compound nuclei formed in the fusion of $_{20}^{40}\mathrm{Ca}+_{28}^{58,64}\mathrm{Ni}$ reactions is investigated by using DCM. The calculations are done for quadrupole $({\ensuremath{\beta}}_{2})$ deformed fragments having optimum orientations for hot configurations. The experimental data for evaporation residues lying within the wide range of center of mass energy $({E}_{\mathrm{c}.\mathrm{m}.})$ of $64--88$ MeV is nicely addressed using the collective clusterization approach of the DCM. The comparative analysis of decay profiles of $_{48}^{98,104}\mathrm{Cd}^{*}$ is worked out by introducing angular momentum and temperature effects in the fragmentation potential and preformation factor.

We have analyzed the importance of the inelastic surface vibrations of colliding nuclei in the sub-barrier fusion enhancement of S1428i+Z4090r system by using the energy dependent Woods-Saxon potential model (EDWSP model) in conjunction with one dimensional Wong formula and the coupled channel formulation using the code CCFULL. The multi-phonon vibrational states of colliding nuclei seem to impart significant contribution. The coupling between relative motion of reactants and these relevant channels in turn produce anomalously large sub-barrier fusion enhancement over the expectations of one dimensional barrier penetration model. Furthermore, the effects of coupling to inelastic surface excitations are imitated due to energy dependence in the Woods-Saxon potential. In EDWSP model calculations, a wide range of diffuseness parameter much larger than the elastic scattering predictions is needed to account the observed fusion enhancement in the close vicinity of Coulomb barrier.

The fusion dynamics of [Formula: see text] reaction has been studied using different theoretical approaches like energy-dependent Woods–Saxon potential (EDWSP) model, coupled channel formulation and Wong approach. At sub-barrier energies, the anomalously large enhancement of the fusion cross-section signifies the importance of barrier modification effects for the adequate addressal of experimental data. The EDWSP model, wherein barrier modification effects are introduced via the energy-dependent diffuseness parameter, is used to examine the sub-barrier fusion anomalies. In the framework of coupled channel model, the impacts of collective excitations and/or static deformations of colliding partners are incorporated in the fusion dynamics. In Wong formula, the role of different Skyrme forces such as SIII, KDE0v1, SkT1, SSk, GSkI is analyzed to address the observed fusion enhancement around the Coulomb barrier. Among these, GSkI and SSk forces seem more appropriate for the addressal of fusion dynamics at sub-barrier energies while SIII, SkT1 and KDE0v1 forces give relatively better results at the above barrier region. The SSk (GSkI) force at higher energies overestimate the experimental data and hence treated with the [Formula: see text]-summed Wong approach. The effect of deformations and optimum orientations is duly incorporated in the calculation and hence gives better description to the observed data. In addition, the fusion cross-sections are predicted over extreme energies using EDWSP and [Formula: see text]-summed Wong approach. It is worth mentioning here that the different theoretical approaches (EDWSP, coupled channel and Wong) induce similar kinds of barrier lowering effects, henceforth, they reasonably describe the sub-barrier fusion data of [Formula: see text] reaction.

This work systematically analyzed the fusion dynamics of the projectile-target combinations involving stable and loosely bound systems within the view of the energy-dependent Woods–Saxon potential model (EDWSP model) and the coupled channel approach. The different projectiles are bombarded onto series of Sm-isotopes, which possess the dominance of the different kinds of the nuclear structure degrees of freedom and with the increase of the neutron richness, the Sm-isotopes gradually shift from spherical shape to a statically deformed shape. In the fusion of [Formula: see text] reaction, the impacts of vibrational degrees of freedom of the colliding nuclei are dominant while in the case of [Formula: see text] systems, the rotational states of the deformed target isotopes have a strong impression on the below-barrier fusion data. The heavier target isotopes ([Formula: see text] also exhibit the higher order deformation such as [Formula: see text], [Formula: see text]-deformation parameter in its ground state and couplings to such channels must be incorporated in theoretical calculations in order to achieve close agreement with the sub-barrier fusion data. However, in the case of the loosely bound systems, the projectile breakup channel significantly affects the fusion excitation functions in the domain of the Coulomb barrier. To ensure the role of the projectile breakup channel, the fusion of the different loosely bound projectiles ([Formula: see text] and [Formula: see text] with Sm-isotopes are investigated, wherein the above-barrier fusion data of these reactions are suppressed with reference to the coupled channel calculations. This hindrance is the result of the projectile breakup effects that occur as a consequence of the breakup of the projectile before reaching the fusion barrier due to its low binding energy. However, in the EDWSP model calculations the magnitude of the hindrance of the above-barrier fusion data of [Formula: see text] and [Formula: see text] reactions is reduced by a factor varying from 7% to 13% with respect to a value reported in the literature. In contrast to this, the sub-barrier fusion enhancement of [Formula: see text] and [Formula: see text] reactions is the result of the dominance of the nuclear structure degrees of freedom of the colliding systems.

This article analyzes the fusion dynamics of loosely bound and stable projectiles with Zr-target isotopes within the context of the coupled channel approach and the energy-dependent Woods-Saxon potential model (EDWSP model). In the case of the 28Si + 90Zr reaction, the coupling to the inelastic surface excitations results in an adequate description of the observed fusion dynamics while in case of the 28Si + 94Zr reaction, the coupling to collective surface vibrational states as well as the neutron (multi-neutron) transfer channel is necessary in the coupled channel calculations to reproduce the below-barrier fusion data. However, the EDWSP model calculation provides an accurate explanation of the fusion data of 28Si + 90,94Zr reactions in the domain of the Coulomb barrier. In the fusion of the 6Li + 90Zr reaction, the inclusion of the nuclear structure degrees of freedom recovers the observed sub-barrier fusion enhancement but results in suppression of the above barrier fusion data by 34% with respect to the coupled channel calculations. Using EDWSP model calculations, this suppression factor is reduced by 14% and consequently, the above-barrier fusion data of 6Li + 90Zr reaction is suppressed by 20% with reference to the EDWSP model calculations. Such fusion suppression at above-barrier energies can be correlated with the breakup of the projectile (6Li) before reaching the fusion barrier, as a consequence of low binding energy.

This article analyzed the fusion dynamics of stable nuclei, loosely bound nuclei and halo nuclei within the view of coupled channel approach and the energy dependent Woods–Saxon potential model (EDWSP model). The different projectiles (${}_{3}{}^{\mathrm{6,7}}{\rm{L}}{\rm{i}}$ and ${}_{8}{}^{\mathrm{16,17}}{\rm{O}})$ are bombarded onto common spherical target isotope (${}_{62}{}^{144}{\rm{S}}{\rm{m}})$ wherein target isotope exhibits inelastic surface excitations as dominant mode of coupling. The fusion dynamics of ${}_{8}{}^{\mathrm{16,17}}{\rm{O}}+{}_{62}{}^{144}{\rm{S}}{\rm{m}}$ reactions, wherein there is no signature of fusion suppression at above barrier energies, are adequately explained by both theoretical approaches in domain of Coulomb barrier. In case of ${}_{3}{}^{\mathrm{6,7}}{\rm{L}}{\rm{i}}+{}_{62}{}^{144}{\rm{S}}{\rm{m}}$ reactions, the couplings of inelastic surface excitations of colliding systems remove the discrepancies between below barrier fusion data and expectations of one-dimensional barrier penetration model. However, at above barrier energies, the fusion excitation function data are suppressed in comparison to the theoretical predictions of both approaches. Interestingly, this suppression of above barrier fusion data is minimized by the present model calculation by a factor of 7% with respect to a value reported in literature. Within the view of the EDWSP model calculations, the above barrier fusion data of ${}_{3}{}^{6}{\rm{L}}{\rm{i}}+{}_{62}{}^{144}{\rm{S}}{\rm{m}}$ reaction are hindered by a factor of 25% and by a factor of 17% for ${}_{3}{}^{7}{\rm{L}}{\rm{i}}+{}_{62}{}^{144}{\rm{S}}{\rm{m}}$ reaction which is smaller than a value as pointed out in literature. The observed sub-barrier fusion dynamics of ${}_{3}{}^{\mathrm{6,7}}{\rm{L}}{\rm{i}}+{}_{62}{}^{144}{\rm{S}}{\rm{m}}$ and ${}_{8}{}^{\mathrm{16,17}}{\rm{O}}+{}_{62}{}^{144}{\rm{S}}{\rm{m}}$ reactions are reasonably described by the coupled channel model and the EDWSP model. Furthermore, the applicability of the EDWSP model has been tested for exploration of the fusion of ${}_{2}{}^{6}{\rm{H}}{\rm{e}}-{\rm{induced}}$ reaction and hence due to barrier modification effects, the EDWSP model successfully explains the observed fusion enhancement of ${}_{2}{}^{6}{\rm{H}}{\rm{e}}+{}_{83}{}^{209}{\rm{B}}{\rm{i}}$ reaction. The present calculations clearly indicate that different theoretical approaches (coupled channel formulation and EDWSP model) induce similar kinds of barrier modification effects in the heavy ion fusion reactions.

The static and energy-dependent nucleus-nucleus potentials are simultaneously used along with the Wong formula for exploration of fusion dynamics of reactions. The role of internal structure degrees of freedom of colliding pairs, such as inelastic surface vibrations, are examined within the context of coupled channel calculations performed using the code CCFULL. Theoretical calculations based on the static Woods-Saxon potential along with the one-dimensional Wong formula fail to address the fusion data of reactions. Such discrepancies can be removed if one uses couplings to internal structure degrees of freedom of colliding nuclei. However, the energy-dependent Woods-Saxon potential model (EDWSP model) accurately describes the sub-barrier fusion enhancement of reactions. Therefore, in sub-barrier fusion dynamics, energy dependence in the nucleus-nucleus potential governs barrier modification effects in a closely similar way to that of the coupled channel approach.

This work deeply analyzed the relative importance of the neutron transfer channels and inelastic surface vibrations of colliding nuclei in the sub-barrier fusion enhancement of various heavy ion systems using an energy dependent Woods–Saxon potential (EDWSP) model in conjunction with a one-dimensional Wong formula and the coupled channel formulation using the code CCFULL. The multi-phonon vibrational states of colliding nuclei and the nucleon transfer channels are found to be dominant internal degrees of freedom. The coupling between the relative motion of reactants and these relevant channels produces anomalously large sub-barrier fusion enhancement over the expectations of the one-dimensional barrier penetration model. In some cases, the influence of neutron transfer dominates over the couplings to low lying surface vibrational states of collision partners. Furthermore, the effects of coupling to inelastic surface excitations and the impact of neutron transfer channels with positive ground state Q-values are imitated due to energy dependence in the Woods–Saxon potential. In the EDWSP model calculations, a wide range for the diffuseness parameter, which is much larger than the value extracted from the elastic scattering data, is needed to account for the observed fusion enhancement in the close vicinity of the Coulomb barrier.