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This paper explores an analytical approach for obtaining multiple solutions for a three-wave interaction system in ( 1 + 1 ) dimensions. We introduce a novel approach that expresses wave solutions in terms of Jacobi elliptic functions and explores specific cases involving hyperbolic functions. Additionally, this paper focuses on analyzing the linear stability of two kinds of solutions: (a) periodic and (b) one or two-hump bright solitons influenced by group velocity and group velocity dispersion. The method of separation of variables along with the ansatz method is employed to derive extract analytical solutions of this model. For linear stability analysis, the eigenvalue problem is solved using the Fourier collocation method, where Fourier coefficients are defined analytically and validated numerically. Moreover, linear stability is verified through direct numerical simulations using the pseudospectral method with special derivatives in the temporal direction ( t ) and the 4th-order Runge–Kutta method in the spatial direction ( z ), further confirmed by the Crank-Nicholson finite difference method. All these investigations within the framework of our current model yield novel insights and present breakthrough research opportunities in the realm of nonlinear optics. A key finding of this study is the discovery of stable analytical solutions, which are presented here for the first time. Furthermore, we introduce a special case known as constant magnitude wave solution and examine its modulational instability in the presence of group velocity dispersion. We also investigate the influence of group velocities and wave vector mismatches. All the results obtained are new and interesting, and the concept opens new possibilities for results in the field of nonlinear optics and nonlinear dynamics.

Exact stationary solutions of nonlinear Schrödinger equation in the presence of complex deformed supersymmetric potential have been obtained in terms of bright soliton and dark soliton. As an example, PT -symmetric Scarf potential has been considered. Then the corresponding spectrum of linear Schrödinger equation has been investigated, and the PT broken and unbroken regions of linear Schrödinger equation have been delineated analytically. The bright soliton and a bright-dark soliton solutions of the nonlinear Schrödinger equation are retrieved analytically with real eigenvalues. Moreover, the stability of these solutions is corroborated by means of linear stability analysis which are validated by direct numerical simulations in terms of a wide range of potential amplitudes for focusing as well as defocusing cases. Finally, we illustrate the strength of stability of bright and dark solitons through the adiabatic transformations on system parameters. Then connected and disconnected stable regions of bright and dark solitons are examined.

For the Deng-Fan potential within a moving boundary condition, the time-dependent Schrödinger equation is considered analytically. The eigenvalue equation is solved by using a combination of Pekeris and Greene-Aldrich approximations. Various time-dependent quantities including density distribution function, auto-correlation function, disequilibrium, average energy, quantum similarity, and quantum similarity index are obtained for selected eight diatomic molecules. The motion of the peak of the density function, with moving boundary condition is investigated for ground states of some diatomic molecules along with the corresponding peak values.

This paper will define the exact solutions of the Schrödinger of a confined particle in an infinite spherical well in both position and momentum spaces. In the present work, we will outline, from an analytical manner in position space and numerically in momentum space for the information-theoretic spreading measures of the wave functions, such as expectation values, root mean square, entropic moment, Shannon entropy, Rényi entropy, and Fisher information. Then we will compare radial density functions in both spaces for these information measures, which are essential from the quantum mechanical point of view.

The time-dependent Schrödinger equation for a particle in a radically confined quantum harmonic oscillator is considered analytically under the influence of a moving boundary condition. Two distinct special cases corresponding to (a) uniformly varying radius and (b) parabolic radius are discussed. These offer solutions in terms of confluent hyper-geometric functions. Beside, the periodic special case is also considered, in which case, the solutions are obtained in terms of Bessel functions. Quantities such as expectation values as well as time-dependent radial density distribution function, time-dependent average energy are obtained in terms of radius of the spherical impenetrable box, in each case. These are considered here for the first time. In addition the Shannon entropy of the radial density functions are calculated quasi exactly.

In this paper, we define three sets of exact solutions for potentials of the type two-parameter Kratzer, inverse square, and Coulomb. The Laguerre polynomials express two sets of solutions, and Bessel functions enter the third one. Also, for these three solutions, we define the exact analytical expressions of the quantum similarities, disequilibria, and entropic moments. In addition, we studied the behavior of quantum dissimilarity, quantum similarity index, and Rényi complexity ratio using every pair of different solutions.

Energy spectrum as well as various information theoretic measures are considered for Hulthén potential in D dimension. For a given ℓ ≠ 0 state, approximate closed expressions are derived, following a simple intuitive approximation for accurate representation of centrifugal term. This is derived from a linear combination of two widely used Greene–Aldrich and Pekeris-type approximations. Energy, wave function, normalization constant, expectation value in r and p space, Heisenberg uncertainty relation, entropic moment of order α ¯ , Shannon entropy, Rényi entropy, disequilibrium, majorization as well as four selected complexity measures like LMC (López-Ruiz, Mancini, Calbert), shape Rényi complexity, Generalized Rényi complexity and Rényi complexity ratio are offered for different screening parameters ( δ ). The effective potential is described quite satisfactorily throughout the whole domain. Obtained results are compared with theoretical energies available in literature, which shows excellent agreement. Performance of six different approximations to centrifugal term is critically discussed. An approximate analytical expression for critical screening for a specific state in arbitrary dimension is offered. Additionally, some inter-dimensional degeneracy occurring in two states, at different dimension for a particular δ is also uncovered.

The present paper describes a quantum mechanical study of Cooper pairs from the point of view of Fermion pair correct spin. The basic idea of building the theory consists of describing each particle with an appropriate Gaussian function, transforming them into a soft quantum mechanical object. From here, a new orthonormalized basis set with adequate symmetry is constructed, and then one can easily build the two-particle spin functions and the two-particle energies. As the particle energies become repulsive, one can add a harmonic oscillator term to stabilize the initial Hamiltonian expectation values. Results indicate that a way to extend the description of Cooper pairs to N Fermion particles becomes a straightforward consequence of the theory developed here.

Two physically important potentials (Manning–Rosen and Pöschl-Teller) are considered for the ro-vibrational energy in diatomic molecules. An improved new approximation is invoked for the centrifugal term, which is then used for their solution within the Nikiforov–Uvarov framework. This employs a recently proposed scheme, which combines the two widely used Greene–Aldrich and Pekeris-type approximations. Thus, approximate analytical expressions are derived for eigenvalues and eigenfunctions. The energies are examined with respect to two approximation parameters, λ and ν . The original approximations are recovered for certain special values of these two parameters. This offers a simple effective scheme for these and other relevant potentials in quantum mechanics.