Exact Two-soliton Solution Research Articles

Numerical Experiments on Extreme Waves Through Oblique–Soliton Interactions

Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev–Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney–Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult-to-attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. Herein, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude.

The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

Electric double layers with surface charge modulations: Exact Poisson-Boltzmann solutions.

Poisson-Boltzmann theory is the cornerstone for soft matter electrostatics. We provide exact analytical solutions to this nonlinear mean-field approach for the diffuse layer of ions in the vicinity of a planar or a cylindrical macroion. While previously known solutions are for homogeneously charged objects, the cases worked out exhibit a modulated surface charge-or equivalently, surface potential-on the macroion (wall) surface. In addition to asymptotic features at large distances from the wall, attention is paid to the fate of the contact theorem, relating the contact density of ions to the local wall charge density. For salt-free systems (counterions only), we make use of results pertaining to the two-dimensional Liouville equation, supplemented by an inverse approach. When salt is present, we invoke the exact two-soliton solution to the 2D sinh-Gordon equation. This leads to inhomogeneous charge patterns, that are either localized or periodic in space. Without salt, the electrostatic signature of a charge pattern on the macroion fades exponentially with distance for a planar macroion, while it decays as an inverse power law for a cylindrical macroion. With salt, our study is limited to the planar geometry and reveals that pattern screening is exponential.

Edge States and Chiral Solitons in Topological Hall and Chern–Simons Fields

The multi-component extension problem of the (2+1) D -gauge topological Jackiw–Pi model describing the nonlinear quantum dynamics of charged particles in multi-layer Hall systems is considered. By applying the dimensional reduction (2 + 1) D → (1 + 1) D to Lagrangians with the Chern–Simons topologic fields , multi-component nonlinear Schrodinger equations for particles are constructed with allowance for their interaction. With Hirota‘s method, an exact two-soliton solution is obtained, which is of interest in quantum information transmission systems due to the stability of their propagation. An asymptotic analysis t →±∞ of soliton-soliton interactions shows that there is no backscattering processes. We identify these solutions with the edge (topological protected) states – chiral solitons – in the multi-layer quantum Hall systems. By applying the Hirota bilinear operator algebra and a current theorem, it is shown that, in contrast to the usual vector solitons, the dynamics of new solutions (chiral vector solitons) has exclusively unidirectional motion. The article is published in the author’s wording.

Exact two-soliton solutions and two-periodic solutions of the perturbed mKdV equation with variable coefficients

We discuss the Darboux transformation method for a modified Korteweg–de Vries equation with variable coefficients and perturbing terms in detail based on the general form of the Darboux transformations for some nonlinear evolution equations solvable by the Ablowitz–Kaup–Newell–Segur inverse scattering method. We use this method to generate families of two-soliton solutions and two-periodic solutions.

Pfaffian solutions and extended Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation

Based on the Pfaffian derivative formula and Hirota bilinear method, the Pfaffian solutions to (3+1)-dimensional Jimbo–Miwa equation are obtained under a set of linear partial differential condition. Moreover, we extend the linear partial differential condition and proved that (3+1)-dimensional Jimbo–Miwa equation has extended Pfaffian solutions. As examples, special exact two-soliton solution and three-soliton solution are computed and plotted. Our results show that (3+1)-dimensional Jimbo–Miwa equation has Pfaffian solutions like BKP equation.

Stability and dynamics of two-soliton molecules

The problem of soliton-soliton force is revisited. From the exact two-soliton solution of a nonautonomous Gross-Pitaevskii equation, we derive a generalized formula for the mutual force between two solitons. The force is given for arbitrary soliton amplitude difference, relative speed, phase, and separation. The latter allows for the investigation of soliton molecule formation, dynamics, and stability. We reveal the role of the time-dependent relative phase between the solitons in binding them in a soliton molecule. We derive its equilibrium bond length, spring constant, frequency, effective mass, and binding energy of the molecule. We investigate the molecule's stability against perturbations such as reflection from surfaces, scattering by barriers, and collisions with other solitons.

Exact Two-Soliton Solutions for Discrete mKdV Equation

An exact two-soliton solution of discrete mKdv equation is derived by using the Hirota direct approach. In addition, we plot the soliton solutions to discuss the properties of solitons. It is worth while noting that we obtain the completely elastic interaction between the two solitons.

Periodic Bifurcation and Soliton Deflexion forKadomtsev–Petviashvili Equation

The spatial–temporal bifurcation for Kadomtsev–Petviashvili (KP)equations is considered. Exact two-soliton solution and doubly periodicsolution to the KP-I equation, and two classes of periodic soliton solutionsin different directions to KP-II are obtained using the bilinearform, homoclinic test technique and temporal and spatial transformationmethod, respectively. The equilibrium solution u0 = −(1/6),a unique spatial–temporal bifurcation which is periodic bifurcation forKP-I and deflexion of soliton for KP-II, is investigated.

EXACT SOLITONS IN THE GROSS–PITAEVSKII EQUATION WITH TIME-MODULATED NONLINEARITY

Exact solitonic solutions of the Gross–Pitaevskii equation with time-modulated nonlinearity of a(t) = a0 / (t + t0) are obtained. With help of these solutions, we analyze the properties of Feshbach-managed solitons in Bose–Einstein condensates in details. Our results show that the parameters of atomic matter waves can be manipulated by proper variation of the scattering length. In particular, an exact two-soliton solution is given, from which, it is shown that the separation between the neighboring solitons can be effectively maintained by allowing the solitons to have unequal initial amplitudes.

Higher-Dimensional KdV Equations and Their SolitonSolutions

A (2+1)-dimensional KdV equation is obtained by use of Hirotamethod, which possesses N-soliton solution, specially its exacttwo-soliton solution is presented. By employing a proper algebraictransformation and the Riccati equation, a type of bell-shapesoliton solutions are produced via regarding the variable in theRiccati equation as the independent variable. Finally, we extendthe above (2+1)-dimensional KdV equation into (3+1)-dimensionalequation, the two-soliton solutions are given.

A certain critical two-soliton solution of the nonlinear Schr dinger equation

An exact two-soliton solution of the nonlinear Schrodinger equation is derived by using the Hirota direct approach. This solution describes such a critical process that two still solitons separated infinitely approach and then pass through each other and keep straight on infinitely.

Chaotic character of two-soliton collisions in the weakly perturbed nonlinear Schrödinger equation.

We analyze the exact two-soliton solution to the unperturbed nonlinear Schrödinger equation and predict that in a weakly perturbed system (i) soliton collisions can be strongly inelastic, (ii) inelastic collisions are of almost nonradiating type, (iii) results of a collision are extremely sensitive to the relative phase of solitons, and (iv) the effect is independent on the particular type of perturbation. In the numerical study we consider two different types of perturbation and confirm the predictions. We also show that this effect is a reason for chaotic soliton scattering. For applications, where the inelasticity of collision, induced by a weak perturbation, is undesirable, we propose a method of compensating it by perturbation of another type.

Large variations in NLS bi-soliton wave groups

The nonlinear Schrodinger (NLS) equation describes the spatial–temporal evolution of the complex amplitude of wave groups in beams and pulses in both second and third order nonlinear material. In this paper we investigate in detail the wave group that has the exact two-soliton solution as amplitude, and show that large variations in the amplitude appear to form a pattern that, at the peak interaction, resembles quite well the linear superposition. The complexity of the phenomenon is a combination of nonlinear effects and linear interference of the carrier waves: the characteristic parameter is the quotient of wave amplitude and frequency difference of the carrier waves, which is also proportional to the quotient of the modulation period of the carrier waves during interaction and the interaction period of the soliton envelopes.

Generation of soliton pairs in nonlinear media with weak dissipation

Weak interaction processes are studied for two KdV solitons at low viscosity. A model is suggested for describing the interaction in terms of slowly varying parameters of the exact two-soliton solution to the KdV equation under perturbation. It is shown that both the inverse scattering problem method and the Witham method lead to the same system of reduced equations. The resulting solutions are in good agreement with numerical calculation data. The main effect is the formation of a bound quasistationary soliton pair and its subsequent dissipation. It turns out that although both the process of soliton merger and the process of dissipation are due to low viscosity, the former is substantially faster.

A Two-Soliton Solution of Cubic Schrödinger Equation

An exact two-soliton solution of cubic Schrödinger equation is derived by using the inverse scattering method, where the transmission coefficient has one pole of second order instead of two simple poles. This solution describes such a process that two infinitely separated solitons approach and then pass through each other and keep straight on infinitely.

Exact two-soliton solutions to the Einstein gravitational field equations

Following our previous work on the existence of 1-soliton solution to the Einstein gravitational field equations in the presence of a spherically-symmetric static background field, we have found six sets of analytical 2-soliton solutions to the Einstein field equations under a certain ansatz in the absence of the stated background field. Numerical analysis shows that if the two solitons of the transverse nature are injected at space variable z→±∞, the longitudinal field componentg 33 will acquire non-zero values for a bounded spatial region at later time. The nature of the solitons becomes rather complex when they interact. The amplitudeg μν of each soliton may change its magnitude resulting from the interaction. We have found that we might interpret the evolution of one field component as the gravitational instanton in our solutions. We must remark also that the total energy of the interacting solitons remains constant, as expected, at all time. These solutions correspond to the situation where the Riemann tensor is in general non-zero and are truly non-trivial solutions.

Soliton and antisoliton resonant interactions

Using the Hirota formalism, Gibbon et al (1976), have shown that the evolution equation ut+ux-uxxt+(4u2+2yxzt)x=0 with u=yt=zx, has the same solitary wave as the regularised long wave (RLW) equation ut+ux-uxxt+6(u2)x=0 and an exact two-soliton solution describing the elastic collision of two sech2 profile solitary waves. Performing a more detailed analysis the authors show that the two-soliton solution can also represent other processes like the resonant or the singular collision of two RLW-type solitary waves. The interaction type depends on the values of a characteristic parameter of the solution. They also prove that with the bilinear form associated with the evolution equation, a three-soliton solution of the Hirota type cannot exist. They then study the equation ut+ux-uxxt+3(u2)x+6utzx=0 with u=zt, associated with another bilinear form, which has the same solitary wave as the evolution equation. They prove the existence of N-soliton solutions, for arbitrary N, and analyse the behaviour of the solitonic solutions. As in the first case, the two-soliton solution can describe elastic, resonant or singular interaction of two RLW-type solitary waves. A remarkable feature of the resonant triad is that it always involves one positive and two negative waves. This triad corresponds to a fundamental vertex for the analysis of the elastic soliton-antisoliton interaction.

The Korteweg-de Vries Two-Soliton Solution as Interacting Two Single Solitons

The exact two-soliton solution of the Korteweg-de Vries (KdV) equation u is decomposed into a simple sum u, + U2, which can be regarded as attractive scattering of two single solitons according to this decomposition. Each Ui satisfies corresponding Interacting KdV equation which is physically natural extension of the original KdV equation. Nonlinear phenomena have been studied exten­ sively for these two decades since the discovery of soliton by Zabusky and Kruskal, ') but still no definite interpretation seems to exist about the interaction of solitons, that is, whether it is at­ tractive or repulsive. This is because N -soliton solution U(N) has been studied as a whole and no detailed analysis has been done so far. We can construct a picture that several single solitons initially apart in space come close, inter­ act with each other satisfying certain coupled nonlinear differential equations and eventually become apart again in space without changing their identity. In our general theory, we will introduce some new operators and decompose the exact N -soliton solution U(N) into N parts, and examine each term to see how it behaves and what equation it satisfies_ They interact always attractively according to our decomposition regardless of the height ratios. But in this paper we concentrate our attention on the simplest N = 2 case, which is very interesting and important. We consider here the Korteweg-de Vries (KdV) equation for u(x, t) in the following form:

A modified regularized long-wave equation with an exact two-soliton solution

Numerical studies of the regularized long-wave (RLW) equation ut+ux+(6u2-uxt)x=0 suggests it has a two-soliton solution although an analytic form for this has not yet been found. The authors show that a modified form of the RLW equation ut+ux+(4u2+2wxvt-uxt)x=0, with u=wt=vx, has an exact two-soliton solution. The modified equation has the same solitary-wave solution as the original equation and its analytic two-soliton solution agrees closely with the numerical solution of the RLW equation.