One-soliton Solutions Research Articles

Vector single- and double-hump solitons for the coupled Sasa-Satsuma equations in optical media

In this paper, the coupled Sasa-Satsuma equations are investigated for the propagation of the ultrashort pulse in such optical media as multi-mode and birefringent fibers. With the Hirota method and introduction of three auxiliary functions, a different bilinear form is obtained. Vector one- and two-soliton solutions in the bright-bright form are constructed. Based on the vector one-soliton solutions, parametric conditions for the existence of the vector single- and double-hump solitons are presented. From the vector two-soliton solutions, elastic and inelastic interactions can both be observed between the (i) two vector single-hump solitons, (ii) two vector double-hump solitons, and (iii) vector single- and double-hump solitons.

Solitons and modulation instability for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensate

Under investigation in this paper are the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensate. We derive the one- and two-soliton solutions for the one, two and three components of the spinor Bose-Einstein condensate. We also discuss the collisions between the two solitons for one and two components of the spinor Bose-Einstein condensate, and find that the collisions are elastic. We make the linear-stability analysis of the solitons for the one and two components of the spinor Bose-Einstein condensate and obtain the conditions for the modulational instability of the solitons, respectively. Based on the one-soliton solutions for the three components of the spinor Bose-Einstein condensate, we see that the period of the one soliton depends on the frequency in space coordinates x axis, ωx, Planck constant, ℏ, and s-wave scattering lengths of the total spin f=2, a2. The smaller ωx and ℏ, the longer the period; the larger a2, the longer the period.

KdV hierarchy with time-dependent coefficients: Lax integrability, bilinear Bäcklund transformation and soliton solutions

Starting from the linear problems, we first derive a new Lax integrable Korteweg-de Vries (KdV) hierarchy with time-dependent coefficients. Then a bilinear Bäcklund transformation (BT) of the variable-coefficient KdV (vcKdV) equation contained in the KdV hierarchy is given. Based on the given bilinear BT, one-soliton solution, two-soliton solution and three-soliton solution are obtained. From these obtained soliton solutions, a uniform formula of explicit n-soliton solutions of the vcKdV equation is summarized. It is graphically shown that the dynamical evolutions of such soliton solutions with time-dependent functions of the KdV hierarchy possess time-varying speeds in the process of propagations.

Integrability, solitons, periodic and travelling waves of a generalized (3+1)-dimensional variable-coefficient nonlinear-wave equation in liquid with gas bubbles

Under investigation in this paper is a generalized (3+1)-dimensional varible-coefficient nonlinear-wave equation, which has been presented for nonlinear waves in liquid with gas bubbles. The bilinear form, Backlund transformation, Lax pair and infinitely-many conservation laws are obtained via the binary Bell polynomials. One-, two- and three-soliton solutions are generated by virtue of the Hirota method. Travelling-wave solutions are derived with the aid of the polynomial expansion method. The one-periodic wave solutions are constructed by the Hirota-Riemann method. Discussions among the soliton, periodic- and travelling-wave solutions are presented: I) the soliton velocities are related to the variable coefficients, while the soliton amplitudes are unaffected; II) the interaction between the solitons is elastic; III) there are three cases of the travelling-wave solutions, i.e., the triangle-type periodical, bell-type and soliton-type travelling-wave solutions, while we notice that bell-type travelling-wave solutions can be converted into one-soliton solutions via taking suitable parameters; IV) the one-periodic waves approach to the solitary waves under some conditions and can be viewed as a superposition of overlapping solitary waves, placed one period apart.

Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the (2+1)-dimensional Sawada–Kotera (SK) equation and (3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota’s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.

Soliton solutions and dynamical evolutions of a generalized AKNS system in the framework of inverse scattering transform

It is important and interesting to derive integrable systems or construct soliton solutions of nonlinear partial differential equations (PDEs). In this paper, a new and more general Ablowitz–Kaup–Newell–Segur (AKNS) system with Lax integrability is derived and solved in the framework of inverse scattering transform (IST). To be specific, the known AKNS’s linear spectral problem and its time evolution equation are first generalized by embedding a nonisospectral parameter whose varying with time obeys the rational function of spectral parameter. Based on the generalized AKNS spectral problem and time evolution equation, we then derive a generalized AKNS system with infinite number of terms. Finally, in the case of reflectionless potentials, explicit n-soliton solutions of the generalized AKNS system are formulated through the IST method. It is shown that the dynamical evolutions of the obtained one-soliton solutions and two-soliton solutions possess time-varying speeds and amplitudes in the process of propagations.

New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Searching for integrable systems and constructing their exact solutions are of both theoretical and practical value. In this paper, Ablowitz–Kaup–Newell–Segur (AKNS) spectral problem and its time evolution equation are first generalized by embedding a new spectral parameter. Based on the generalized AKNS spectral problem and its time evolution equation, Lax integrability of a nonisospectral integrodifferential system is then verified. Furthermore, exact solutions of the nonisospectral integrodifferential system are formulated through the inverse scattering transform (IST) method. Finally, in the case of reflectionless potentials, the obtained exact solutions are reduced ton-soliton solutions. Whenn=1andn=2, the characteristics of soliton dynamics of one-soliton solutions and two-soliton solutions are analyzed with the help of figures.

Soliton-like, periodic wave and rational solutions for a (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation in the incompressible fluid

Under investigation in this paper is a (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation in the incompressible fluid. With the aid of the bilinear form, Nth-order soliton-like solutions are obtained via the Pffafian method, rational solutions are derived with the ansätz method and periodic wave solutions are constructed via the Riemann theta function. The analytic solutions obtained via the Pffafian method are similar to the kink solitons, while, the interaction regions with little peaks are different from those of the usual kink solitons. The rational solutions which have one upper lump and one down deep hole are the bright-dark solitary wave solutions. For the rational solutions which combine the kink solitary wave with breather-like wave, asymptotic behaviors show that the breather-like wave disappears with the evolution of t. Relations between the one-soliton solutions and one-periodic wave solutions are analysed, which exhibit the asymptotic behaviors of the periodic waves.

Bilinear Bäcklund transformation, soliton and periodic wave solutions for a $$\varvec{(3 + 1)}$$ ( 3 + 1 ) -dimensional variable-coefficient generalized shallow water wave equation

Under investigation in this paper is a \((3 + 1)\)-dimensional variable-coefficient generalized shallow water wave equation. Bilinear forms, Backlund transformation and Lax pair are obtained based on the Bell polynomials and symbolic computation. One-, two- and three-soliton solutions are derived via the Hirota method. One-periodic wave solutions are obtained via the Hirota–Riemann method. Discussions indicate that the one-periodic wave solutions approach to the one-soliton solutions when \(\varTheta \rightarrow 0\). Propagation and interaction of the soliton solutions have been discussed graphically. We find that not the soliton amplitudes, but the velocities are related to the variable coefficients \(\delta _{1}(t)\) and \(\delta _{2}(t)\). Phase shifts of the two-soliton solutions are the only differences to the superposition of two one-soliton solutions, so the amplitudes of the two-soliton solutions are equal to the sum of the corresponding two one-soliton solutions.

Symmetry-broken canonizations of the semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site coupling

The integrable system of nonlinear Schrödinger type consisting of two mutually symmetric basic subsystems and one concomitant subsystem given on a ribbon of triangular lattice is characterized by the highly nonstandard form of Poisson structure. The fact of system criticality against the positive semi-definite background parameter provides us the clue how to organize the nonlinear point transformation ensuring the canonization of field variables. Precisely it allows to introduce two subsets of mutually symmetric intermediate (auxiliary) field variables and to involve one or another of them into a rational procedure of canonization realizable already in terms of two mutually asymmetric subsets of field amplitudes. There are two variants of system asymmetric standardization (namely, the minus-asymmetric and plus-asymmetric ones) emanated from the two-leg structure of underlying space lattice of the primary (noncanonical) nonlinear system. Either of the two asymmetrically standardized nonlinear systems consists of two canonical subsystems which can be referred to as the strong and the weak ones. The strong subsystem is the subsystem of bright nonlinear excitations on the whole admissible interval of the background parameter. In contrast the type of nonlinear excitations in the weak subsystem depends on the chosen domain of the background parameter. Thus in the under-critical region of the background parameter, the weak subsystem behaves as the subsystem of bright excitations while in the over-critical region it turns into the subsystem of dark excitations. Moreover in the very critical point, the field amplitudes of weak subsystem shrink to zero and the canonical system as the whole is reduced to the single strong subsystem. The general results concerning both minus-asymmetric and plus-asymmetric standardizations are confirmed analytically by means of the system multi-component one-soliton solution.

Regularized degenerate multi-solitons

We report complex PT-symmetric multi-soliton solutions to the Korteweg de-Vries equation that asymptotically contain one-soliton solutions, with each of them possessing the same amount of finite real energy. We demonstrate how these solutions originate from degenerate energy solutions of the Schroedinger equation. Technically this is achieved by the application of Darboux-Crum transformations involving Jordan states with suitable regularizing shifts. Alternatively they may be constructed from a limiting process within the context Hirota's direct method or on a nonlinear superposition obtained from multiple Baecklund transformations. The proposed procedure is completely generic and also applicable to other types of nonlinear integrable systems.

The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: The particular integrable case and soliton solution

In this paper, we investigate the one-dimensional parabolic-parabolic Patlak-Keller-Segel model of chemotaxis. For the case when the diffusion coefficient of chemical substance is equal to two, in terms of travelling wave variables the reduced system appears integrable and allows the analytical solution. We obtain the exact soliton solutions, one of which is exactly the one-soliton solution of the Korteweg-de Vries equation.

Gravitational two solitons in Levi-Cività spacetime

Applying the Pomeransky inverse scattering method to the four-dimensional vacuum Einstein equations and using the Levi-Cività solution as a seed, we construct a two-soliton solution with cylindrical symmetry. In our previous work, we constructed the one-soliton solution with a real pole and showed that the singularities that the Levi-Cività background has on an axis can be removed by the choice of certain special parameters, but it still has unavoidable null singularities, as usual one-solitons do. In this work, we show that for the two-soliton solutions, any singularities can be removed by suitable parameter-setting and such solutions describe the propagation of gravitational wave packets. Moreover, in terms of the two-soliton solutions, we mention a time shift phenomenon, the coalescence and the split of solitons as the nonlinear effect of gravitational waves.

Complex solitons with real energies

Using Hirota’s direct method and Bäcklund transformations we construct explicit complex one and two-soliton solutions to the complex Korteweg–de Vries (KdV) equation, the complex modified KdV (mKdV) equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be -symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex -symmetric solutions to the KdV equation may also be constructed alternatively from real solutions to the mKdV by means of Miura transformations. The multi-soliton solutions obtained from Hirota’s method break the -symmetric, whereas those obtained from Bäcklund transformations are -invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are -symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space–time and the fact that any of our N-soliton solutions may be decomposed into N separate -symmetrizable one-soliton solutions.

An extended modified KdV equation and its Painlevé integrability

In this work we present an extended higher-order modified KdV equation. An analysis is carried out to show that this equation admits the Painleve property. For this new integrable model, the one-soliton, two-soliton and three-soliton solutions are derived by using the simplified Hirota’s direct method. We also demonstrate that one, two and three singular soliton solutions are possible for the defocusing form of this extended higher-order mKdV equation.

Dark solitons and rouge waves for a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential in the Bose–Einstein condensate

Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential, which describes the dynamics of the (2+1)-dimensional Bose–Einstein condensate. Employing the Hirota method and symbolic computation, we obtain the dark one-soliton, two-soliton, three-soliton, breather-wave and rouge-wave solutions, respectively. We graphically study the dark solitons with the time-varying harmonic potential and scaled scattering length. Parallel and period solitons are observed. We obtain that when the external trapping potential increases with time, amplitudes of the dark solitons increase and widths of those solitons become narrower; when the external trapping potential is a periodic function, amplitudes and widths of the dark solitons periodically change. Decrease in the scaled scattering length leads to the narrower solitons’ widths, but does not affect the solitons’ amplitudes. Breather waves and rouge waves are also displayed: Rouge waves emerge when the period of the breather waves go to the infinity.

Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation

It is shown that the nonlocal nonlinear Schr\odinger equation recently proposed by Ablowitz and Musslimani [Phys. Rev. Lett. 110, 064105 (2013)] is gauge equivalent to the unconventional system of coupled Landau-Lifshitz equations. The first integrals of motion and one-soliton solution of an obtained model are given. The physical and geometrical aspects of model and their effect on expected metamagnetic structures are studied.

Analytical study of solitons to nonlinear time fractional parabolic equations

The some of the well-known nonlinear time fractional parabolic partial differential equations is studied in this paper. The fractional complex transform and the first integral method are employed to construct one-soliton solutions of these equations. The power of this manageable method is confirmed. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions.

Soliton model of a photon propagating in dielectrics

The previously proposed three-dimensional soliton model of a photon propagating in vacuum is modified to describe its propagation in a homogeneous linear dielectric medium. The one-soliton solution of the derived nonlinear equation correctly predicts the energy and the Abraham and Minkowski momenta of the photon in dielectrics. A new nonlinear equation is proposed, which has a one-soliton solution that at every point oscillates with the same frequency and falls exponentially in the longitudinal, as well as in the transverse direction from the centre of the soliton.