Search Results (6)

This work extends the analytical operation of the Riemann–RHPHilbert approach (RHA) for fractional-order nonlinear integrable systems under the solvable meaning of inverse scattering transform (IST) to variable-coefficient fractional-order nonlinear models. Firstly, based on the matrix spectral problem proposed by Ablowitz, Kaup, Newell, and Segur, this article derives an integer-order integrable system, which is abbreviated as the AKNS hierarchy. Secondly, by taking specific values of the operator in the derived AKNS hierarchy, a variable-coefficient fractional higher-order NLS hierarchy (vfhNLSH) is obtained, and its anomalous dispersion relation (ADR) is derived via formal solution. Significantly, the reductions of the vfhNLSH include three variable-coefficient fractional-order integrable models: the Hirota equation (vfHE), the Lakshmanan–Porsezian–Daniel equation (vfLPDE), and the fifth-order NLS equation (vffNLSE). Finally, we conduct a detailed study on the representative vfHE as an example rather than a special case and construct its explicit N-fold analytical solution based on the extension of the RHA. At the same time, numerical visualization simulations are conducted to demonstrate the waveform structure characteristics of the solutions under $N=1$ and $N=2$ conditions, including solitons, breathers, and their coupled nonlinear waves. The same process is fully applicable to the other two reduced models, with only some differences in the related results and the dynamic behavior of the solutions. It is shown that the temporal part of the Lax pair associated with the vfHE cannot yet be explicitly determined. Therefore, the fractional-order extension of the RHA presented in this article constitutes a formal or RHA-inspired construction, rather than a fully rigorous fractional-order RHA extension. Full article

Ion-acoustic waves in dense quantum plasmas are strongly influenced by Fermi degeneracy, Landau quantization, and finite-temperature effects, and in many relevant environments, they also experience memory and nonlocal transport processes that cannot be captured within the planar integer Korteweg-de Vries (KdV) paradigm. In the present work, we revisit this problem by considering a two-fluid, partially degenerate electron-ion plasma in which electron trapping in the presence of a quantizing field and finite temperature is taken into account. Starting from the normalized fluid-Poisson system appropriate for such magnetized quantum plasmas, the reductive perturbation technique is used to derive the planar integer KdV equation for weakly nonlinear ion-acoustic disturbances. Within this integer-order KdV framework, we recast the evolution equation as a planar dynamical system, construct the associated Hamiltonian and effective Sagdeev-like potential, and demonstrate the existence of compressive solitary waves and nonlinear periodic modes via homoclinic and periodic phase-space orbits. Exact multi-soliton solutions and interaction states are then obtained by combining Hirota’s direct bilinear method with generalized Wronskian representations, allowing us to describe not only standard one-, two-, and three-soliton profiles but also positon-negaton interactions relevant to magnetized, partially degenerate plasmas. To incorporate hereditary and history-dependent effects that arise from anomalous transport and nonlocal temporal response in dense environments, we extend the model by introducing a Caputo time-fractional derivative, thereby obtaining a time-fractional KdV (FKdV) equation that continuously connects the classical KdV limit to fractional dynamics. The FKdV equation is analyzed using the Tantawy technique. This semi-analytical iterative scheme yields rapidly convergent series approximations for the fractional ion-acoustic soliton and provides explicit control of the approximation error. The fractional solutions show that varying the order of the Caputo derivative modifies the amplitude, width, and temporal relaxation of the solitary structures and can even split the pulse into two distinct lobes, in contrast with the nearly rigid propagation predicted by the integer-order KdV equation. Taken together, these results clarify how Landau quantization, finite electron temperature, and fractional-order memory jointly shape the morphology, robustness, and interaction properties of ion-acoustic structures in strongly magnetized quantum plasmas of astrophysical and high-energy-density laboratory interest. Full article

This study addresses the complete integrability of a generalized multicomponent version of the first multiplicative Bogoyavlensky lattice with branched dispersion. The analysis is performed using the Hirota bilinear formalism and the periodic reduction technique. Initially, a two-dimensional mB1 lattice is considered, for which complete integrability is established by constructing its bilinear form and general multisoliton solutions via the Hirota bilinear formalism. A periodic reduction along the discrete independent variable is then applied to derive the coupled multicomponent mB1 lattice, along with its corresponding bilinear representation and multisoliton solutions. The resulting system serves as an integrable semi-discrete generalization of the classical Volterra-type equation. These findings contribute to the broader understanding of integrable lattice systems with branched dispersion relations and provide a constructive framework for obtaining explicit soliton solutions in multicomponent systems, which exhibit rich internal symmetry structures. Full article

This paper focuses on the variable coefficients geophysical KdV (VCGKdV) equation, which involves time-dependent perturbation, nonlinearity and dispersion parameters. It is a more realistic model than its constant coefficient counterpart and can be useful to, for instance, investigate the Coriolis effect on oceanic flows. Firstly, we analyzed this model using three strong methods that allow the investigation of its integrability: the Lie symmetry approach, Painlevé property and Hirota formalism. The general constraints between the involved parameters under which the complete integrability in Lie, Painlevé or Hirota sense exists, as well as the largest class of this type of equations, which admits the same class of imposed symmetries are generated. Then, some new specific families of solutions for the model endowed with either Lie symmetry properties, Lie and Painlevé constraints or with Lie, Painlevé and Hirota constraints were generated and compared with solutions derived with other techniques. By numerical simulations, the dynamical behaviors of some Lie invariant solutions and nonautonomous multiple solitons are depicted. Full article

We present a theoretical and numerical study of soliton formation and dynamics in a Bose–Einstein condensate (BEC) confined within a symmetric harmonic trap, subjected to an external barrier potential. Our investigation focuses on the role of symmetry in the system, particularly highlighting how the interplay between the harmonic confinement and the barrier shape governs the generation and evolution of multi-soliton states. Employing a reduction of the 3D Gross–Pitaevskii equation to lower-dimensional regimes, we analyze the behavior of dark solitons in 2D and 1D configurations using, for the former, exact solutions constructed from the Hirota’s bilinear formalism. We observe that the number of generated solitons exhibits a plateau-like dependence on the height of the potential barrier, reflecting the system’s symmetry and nonlinearity. Furthermore, we break the central symmetry by translating the barrier, leading to asymmetrical soliton patterns and novel dynamical behaviors. These findings underline the fundamental role of symmetry in the formation and stability of solitons in confined quantum gases, offering new perspectives on soliton engineering in trapped BECs. Full article

We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts. Full article