Multihump Solitons Research Articles

Nondegenerate solitons of the (2+1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optical fibers

We derive the multi-hump nondegenerate solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations with propagation distance dependent diffraction, nonlinearity and gain (loss) using the developing Hirota bilinear method, and analyze the dynamical behaviors of these nondegenerate solitons. The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers, varying diffraction and nonlinearity parameters. In addition, when all the variable coefficients are chosen to be constant, the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons. Finally, it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.

Propagation of coupled quartic and dipole multi-solitons in optical fibers medium with higher-order dispersions

We present the discovery of two types of multiple-hump soliton modes in a highly dispersive optical fiber with a Kerr nonlinearity. We show that multi-hump optical solitons of quartic or dipole types are possible in the fiber system in the presence of higher-order dispersion. Such nonlinear wave packets are very well described by an extended nonlinear Schrödinger equation involving both cubic and quartic dispersion terms. It is found that the third- and fourth-order dispersion effects in the fiber material may lead to the coupling of quartic or dipole solitons into double-, triple-, and multi-humped solitons. We provide the initial conditions for the formation of coupled multi-hump quartic and dipole solitons in the fiber. Numerical results illustrate that propagating multi-quartic and multi-dipole solitons in highly dispersive optical fibers coincide with a high accuracy to our analytical multi-soliton solutions. It is important for applications that described multiple-hump soliton modes are stable to small noise perturbation that was confirmed by numerical simulations. These numerical results confirm that the newly found multi-soliton pulses can be potentially utilized for transmission in optical fibers medium with higher-order dispersions.

Scalar and vector supermode solitons owing to competing nonlocal nonlinearities.

We investigate scalar and vector multi-hump spatial solitons resulting from competing Kerr-like nonlinear responses excited in a nonlocal medium by either one or two (mutually incoherent) light beams. Two-color vector supermode solitons are more amenable to control but exhibit an intriguing form of spontaneous symmetry breaking in propagation.

The structure of algebraic solitons and compactons in the generalized Korteweg–de Vries equation

We analyze the main properties of soliton solutions to the generalized KdV equation ut+[F(u)]x+uxxx=0, where the leading term F(u)∼quα, α>0, q∈R. The far field of such solitons may have three options. For q>0 and α>1 the analysis re-confirmed that all traveling solitons have “light” exponentially decaying tails and propagate to the right. If q<0 and α<1, the traveling solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated Fu and α>1 (e.g., the Gardner equation) standing algebraic solitons with “heavy” power-law tails may appear. If the leading term of Fu is negative, the set of solutions may include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic solitons and compactons with any of the three types of tails. The solutions usually have a single-hump structure but if Fu represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.

Alternative solitons in the Hirota–Satsuma system via the direct method

Multisolitons and multi-hump solitons are constructed for the coupled Korteweg–de Vries (cKdV) system introduced by Hirota and Satsuma (1981) in the context of the propagation of two waves interacting with different dispersion relations. The direct method analytically exhibit a new class of complementary or alternative soliton solutions functionally different to the traditional or classical multisolitons known (Hirota and Satsuma, 1981; Ramani et al., 1983; Parra Prado and Cisneros-Ake, 2020). As in the classical case, our alternative multisolitons are found to obey elastic interactions of the N-KdV and M-cKdV solitons. To show this, we inductively construct tables to describe in detail the mechanism of the different wave interactions and the coefficients involved in them. As a particular scenario, it is found that each of the 1-cKdV and 1-KdV-1cKdV interaction cases may produce two independent criteria for the appearance of one-soliton solutions consisting of two humps in shape. A proper combination of these two conditions gives place to multi-hump solitons. We explicitly show the two-one, two-two, three-one and three-two humps for the two-component wave solutions. • The direct method due to Hirota allows to find complementary two-component multisolitons in a coupled KdV system. • Symmetric one-soliton solutions with two humps in shape are obtained from two independent criteria. • Multi-hump alternative solitons are constructed from two different two-hump conditions.

New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM

In this article, a modification of the rapidly convergent approximation method is proposed to solve a coupled Korteweg–de Vries equations with conformable derivative that govern shallow-water waves. Based on the Leibniz and chain rule of conformable derivative, these equations reduced into ODEs with integer-order using traveling wave transformation. Adopting the modified scheme a new novel exact solution of the reduced coupled ordinary differential equations is obtained in terms of exponential functions. Finally, by putting them back into traveling wave transformation the solutions of the considered partial differential equations with conformable derivative are derived. To ensure the boundedness of the derived solutions few theorems have been proposed and proved. The derived results of the theorems are utilized to plot the solutions. Graphics exhibit that solutions have variant multi-hump soliton peculiarities and their tails decay to zero exponentially in a monotonic manner. These results not only show the efficiency of the modified scheme but also establish that the solution is enriched with new multi-hump features.

Modulated solitons, soliton and vortex clusters in purely nonlinear defocusing media

By means of standard Thomas–Fermi approximation, we present stable solutions of one- and two-dimensional (1D and 2D) multihump soliton structures, including 1D and 2D modulated solitons as well as 2D soliton clusters, supported by inhomogeneous defocusing nonlinear Kerr media. Such nonlinear settings may apply to nonlinear optics and Bose–Einstein condensates. All the 1D and 2D fundamental modes with different numbers of humps (for both modulated solitons and soliton clusters) are demonstrated to be robustly stable. Besides, the considered 2D nonlinear setting also supports a vast varieties of vortex modes—vortex soliton clusters. Such novel type of vortex soliton clusters is qualitatively different from the ordinary vortex clusters, on the account of the fact that the latter type is a single large soliton with several-vortex configurations inside it, while our predicted vortex clusters structures are composed of the arrays of unitary vortex solitons. Stability regions for all the soliton structures are identified via linear stability analysis and direct simulations.

Vector bright solitons and their interactions of the couple Fokas–Lenells system in a birefringent optical fiber

In the fiber communication domain, people are facing the challenges due to the rapidly growing requirement on the capacity from new functions and services. Multi-hump solitons are therefore noticed and studied on the feasibility of improving the capacity of the optical fiber communication. In this paper, we study the vector bright solitons and their interaction properties of the coupled Fokas–Lenells system, which models the femtosecond optical pulses in a birefringent optical fiber. We derive the so-called degenerate and nondegenerate vector solitons associated with the one and two eigenvalues, respectively, and the latter admits the symmetric profile. Asymptotically and graphically, interaction patterns of such solitons are classified as follows: Interactions between the degenerate solitons can be elastic or inelastic, reflecting the intensity redistribution between the two components; Interactions between the degenerate and nondegenerate solitons are inelastic, which make the nondegenerate solitons maintaining or losing the profiles in the different situations; Interactions between the nondegenerate solitons do not cause the intensity redistribution, while their shapes change slightly or remain unchanged.

One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: nonlinear lattice

The existence and stability of stable bright solitons in one-dimensional (1D) media with a spatially periodical modulated Kerr nonlinearity are demonstrated by means of the linear-stability analysis and in direct numerical simulations. The nonlinear potential landscape can balance the fractional-order diffraction and thus stabilizes the solitons, making the model unique and governed by the recently introduced fractional Schr\"{o}dinger equation with a self-focusing cubic nonlinear lattice. Both 1D fundamental and multihump solitons (in forms of dipole and tripole ones) are found, which occupy one or three cells of the nonlinear lattice respectively, depending on the soliton's power (intensity). We find that the profiles of the predicted soliton families are impacted intensely by the L\'{e}vy index $\alpha$ which denotes the level of fractional Laplacian, so does to their stability. The stabilization of soliton families is possible if $\alpha$ exceeds a threshold value, below which the balance between fractional-order diffraction and the spatially modulated focusing nonlinearity will be broken.

Multi-hump bright and dark solitons for the Schrödinger-Korteweg-de Vries coupled system.

The Hirota bilinear method is extended to find one- and two-hump exact bright and dark soliton solutions to a coupled system between the linear Schrödinger and Korteweg-de Vries (KdV) equations arising in the energy transfer problem along a cubic anharmonic crystal medium. The bilinear form associated to this system is found by imitating the well known bilinearizing transformations used in the standard nonlinear Schrödinger (NLS) and KdV equations. Proper finite exponential expansions in the transformed variables allow one to exhibit multihump soliton solutions as single entities resulting from the adjustment of appropriate dispersion relations between the wave parameters describing the profiles. It is found that such one- and two-hump solutions correspond to the one- and two-KdV solitons trapped by both the bright and dark-gray NLS solitons. Our two-hump bright and dark solutions represent novel solutions for the type of interactions and nonlinearities considered.

AN ANALYTICAL SCHEME FOR THE ANALYSIS OF MULTI-HUMP SOLITONS

An analytical framework for the analysis of multi-hump solitons is proposed in this paper. Multi-hump solitons are defined by imposing special symmetry conditions on the classical soliton expression. Such soliton solutions have a wide range of potential applications in the field of optical communications. The proposed algebras of soliton solutions enable a new look at the propagation dynamics of complex nonlinear wave phenomena. The efficiency of the presented analytical scheme is demonstrated using a system of Riccati differential equations with diffusive and multiplicative coupling.

Multi-hump bright solitons in a Schrödinger–mKdV system

We consider the problem of energy transport in a Davydov model along an anharmonic crystal medium obeying quartic longitudinal interactions corresponding to rigid interacting particles. The Zabusky and Kruskal unidirectional continuum limit of the original discrete equations reduces, in the long wave approximation, to a coupled system between the linear Schrödinger (LS) equation and the modified Korteweg–de Vries (mKdV) equation. Single- and two-hump bright soliton solutions for this LS–mKdV system are predicted to exist by variational means and numerically confirmed. The one-hump bright solitons are found to be the anharmonic supersonic analogue of the Davydov's solitons while the two-hump (in both components) bright solitons are found to be a novel type of soliton consisting of a two-soliton solution of mKdV trapped by the wave function associated to the LS equation. This two-hump soliton solution, as a two component solution, represents a new class of polaron solution to be contrasted with the two-soliton interaction phenomena from soliton theory, as revealed by a variational approach and direct numerical results for the two-soliton solution.

Bright nonlocal quadratic solitons induced by boundary confinement

Under the Dirichlet boundary conditions, a family of bright quadratic solitons exists in the regime where the second harmonic can be regarded as the refractive index of the fundamental wave with an oscillatory nonlocal response. By simplifying the governing equations into the Snyder-Mitchell mode, the approximate analytical solutions are obtained. Taking them as the initial guess and using a numerical code, we found two branches of bright solitons, of which the beam width increases (branch I) and decreases (branch II) with the increase of the sample size, respectively. If the nonlocality is fixed and the sample size is varied, the soliton width varies piecewise and approximately periodically. In each period, solitons only exist in a small range of sample size. Single-hump fundamental wave solitons with the same beam width in narrower samples can be, if the second harmonics are connected smoothly, jointed to be a multihump soliton in a wider sample whose size is the sum of those for the narrower ones. The dynamical simulation shows that the found solitons are unstable.

Theory on bright and dark soliton formation in strongly magnetized plasmas

The existence and properties of bright and dark solitons in strongly magnetized warm plasmas are investigated analytically. These solitons are solutions to a four-dimensional Hamiltonian system with zero and non-zero boundary conditions. Based on the dynamical systems theory, the parametric domains of magnetic field and soliton frequency for the existence of bright and dark solitons in both cold and warm plasmas are identified. It is found that the temperature effects play an important role in determining the parametric domains and properties of the solitons. Specifically, the temperature effects make it possible for the existence of multi-hump dark solitons and remove the constraint on magnetic field and soliton frequency caused by the non-negative condition of the density profile. The numerical integration of the soliton equations shows that the bright soliton amplitude increases with magnetic field while the dark soliton amplitude decreases with it. For both bright and dark solitons, the temperature effects suppress the soliton amplitude.

Multihump solitons in two-dimensional parity-time-symmetric optical lattices with focusing saturable nonlinearity

We study the existence and stability of multihump solitons in two-dimensional (2D) parity-time (PT)-symmetric periodic potentials with focusing saturable nonlinearity. All the humps of these solitons are in consecutive lattice sites and the adjacent two humps are out-of-phase. These multihump solitons exist in finite regions in the semi-infinite gap and cannot bifurcate from the edge of the first Bloch band. They can be stable in the moderate power region. It is found that the saturation parameter will affect the existence and stability of these multihump gap solitons significantly. The transverse power flow vector in these 2D multihump solitons in focusing saturable media is also investigated.

Spatial Patterns of Dissipative Polariton Solitons in Semiconductor Microcavities.

We report propagating bound microcavity polariton soliton arrays consisting of multipeak structures either along (x) or perpendicular (y) to the direction of propagation. Soliton arrays of up to five solitons are observed, with the number of solitons controlled by the size and power of the triggering laser pulse. The breakup along the x direction occurs when the effective area of the trigger pulse exceeds the characteristic soliton size determined by polariton-polariton interactions. Narrowing of soliton emission in energy-momentum space indicates phase locking between adjacent solitons, consistent with numerical modeling which predicts stable multihump soliton solutions. In the y direction, the breakup originates from inhomogeneity across the wave front in the transverse direction which develops into a stable array only in the solitonic regime via phase-dependent interactions of propagating fronts.

Effects of next-nearest-neighbor interactions on discrete multibreathers corresponding to Davydov model with saturable nonlinearities

The influence of next-nearest-neighbor interactions (next-NNI) of dipole–dipole type is analyzed in Davydov model with saturable nonlinearities. We analytically study the regions of discrete modulational instability (MI) of plane carrier waves and it appears that this region decreases as the number of nearest neighbors, m, increases. We also show via the instability growth rate (gain) that when m increases, bandwidth of instability decreases. Otherwise, it is noted that the saturation also has an antagonistic effect on the gain. Numerical simulations indicate that the presence of next-NNI induced downward corrections of the time of onset of MI. After having sought Discrete Soliton (DS) and Discrete Multihump soliton (DMHS) numerically with m=1; 2; 3, the next-nearest neighbor dependence of the width and height of these solutions is discussed. A study of mobility is achieved and it results that next-NNI increase the speed of DMHS. Furthermore, the collisions of two DMHS are performed and it emerges mainly that next-NNI lead to the formation of large stationary solitons.

Mobility of discrete multibreathers in the exciton dynamics of the Davydov model with saturable nonlinearities.

We show that the state of amide-I excitations in proteins is modeled by the discrete nonlinear Schrödinger equation with saturable nonlinearities. This is done by extending the Davydov model to take into account the competition between local compression and local dilatation of the lattice, thus leading to the interplay between self-focusing and defocusing saturable nonlinearities. Site-centered (sc) mode and/or bond-centered mode like discrete multihump soliton (DMHS) solutions are found numerically and their stability is analyzed. As a result, we obtained the existence and stability diagrams for all observed types of sc DMHS solutions. We also note that the stability of sc DMHS solutions depends not only on the value of the interpeak separation but also on the number of peaks, while their counterpart having at least one intersite soliton is instable. A study of mobility is achieved and it appears that, depending on the higher-order saturable nonlinearity, DMHS-like mechanism for vibrational energy transport along the protein chain is possible.

Vortex and dipole solitons in lattices possessing defects and dislocations

We report the numerical existence of dipole and vortex solitons for the two-dimensional nonlinear Schrödinger (NLS) equation with external potentials that possess strong irregularities, i.e., edge dislocations and a vacancy defects. Multi-humped solitons are computed by employing a spectral fixed-point computational scheme. The nonlinear stability of these solitons is investigated using direct simulations of the NLS equation and it is observed that these multi-humped modes in the defect lattices can be stable or unstable.

Vector solitons in parity-time-symmetric lattices

I study vector solitons involving two incoherently coupled field components in periodic parity-time (PT)-symmetric optical lattices. The specific symmetry of the lattice imposes restrictions on the symmetry of available vector soliton states. While all configurations with asymmetric intensity distributions are prohibited, such lattices support multihump solitons with an equal number of "in-phase" or "out-of-phase" spots in two components, residing on neighboring lattice channels. In the focusing medium, only the solitons containing out-of-phase spots in at least one component can be stable, while in the defocusing medium stability is achieved for structures consisting of in-phase spots. Mixed-gap vector solitons with components emerging from different gaps in the lattice spectrum also exist and can be stable in the PT-symmetric lattice.