All the above models arise in the description of shallow water waves, and their solutions are used to construct the corresponding soliton solutions of the nonlocal NLS. The reduction to the KP model allows us to construct the approximate soliton solutions of the nonlocal NLS, which are presented in Section 3.
Dark lump solitons
We start with a 2–2 interaction, defining the matrix A, such that. corresponding to relativeθmj4m¼1. Similar to the case of antidark strip solitons, once Eq. 48) gives rise to the approximate dark soliton of the nonlocal NLS.
Ring dark and antidark solitons
This is of the form of Eq. 48), with the amplitude and velocity of the soliton varying ast2=3, and the width varying ast1=3, as follows from Eqs. First, in Figure 9, 3D plots depicting the profiles of dark ring solitons and anti-dark ring solitons are shown.
Conclusions and discussion
Regarding the possibility of the experimental observation of the other soliton states predicted by our analysis, we note the following. The results obtained are based on the formal reduction of the non-local NLS model to the KP (cKP) equation.
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Introduction
Although DNLS comparison is similar to NLS comparison in form, it does not belong to the famous AKNS hierarchy at all. The IST of the DNLS equation is therefore very different from that of the NLS equation known to us.
An N-soliton solution to the DNLS equation based on a revised inverse scattering transform
- The revised inverse scattering transform and the Zakharov-Shabat equation for DNLS equation with VBC
- Explicit expression of N -soliton solution
- The typical examples for one- and two-soliton solutions
- The asymptotic behaviors of N -soliton solution
We give two concrete examples – one- and two-soliton solutions as illustrations of the general explicit soliton solution. Finally, we suggest that the exact N -soliton solution of the DNLS equation can be converted to the MNLS equation by a gauge-like transformation.
A simple method to derive and solve Marchenko equation for DNLS equation
- The lax pair and its Jost functions of DNLS equation
- Marchenko equation for DNLSE and its demonstration
- A multi-soliton solution of the DNLS equation based upon pure Marchenko formalism
- The special examples for one- and two-soliton solutions
Due to the symmetry of the first Lax operator λ2ð iσ3Þ11 ¼λ2ð iσ3Þ22 and λU21 ¼ λU12, the kernel array N x,ð yÞ of the integral representation of the Jost function must have the same symmetry as follows:. We obtain two equations with terms λ2 and λoutside the integralА. dy Sf g ¼0, where gS equals λ2Ndxðx,yÞ þλNoxðx,yÞ. ¼iσ3F0ðzþyÞ (144) Given the dependence of the Jost solutions on the squared spectral parameterλ2, we choose in the reflectionless case.
Thus, the time-independent part is indispensable and can be absorbed or normalized only by redefining the soliton center and initial phase. When there are simple Npolesλ1,λ2,⋯,λIn the first quadrant of the complex plane of λ, the Marchenko equation will give an N-solution to the DNLS equation with VBC in the no-reflection case. Finally, substituting (174) into (177) and (173 and 177) into (162), we thus achieve the clear N-soliton solution of the DNLS equation with VBC in the reflection-free case, based on a pure Marchenko formalism and without the need for the concrete expression of the spectrum of Þ.
Soliton solution of the DNLS equation based on Hirota’s bilinear derivative transform
- Fundamental concepts and general properties of bilinear derivative transform
- Bilinear derivative transform of DNLS equation
- Soliton solution of the DNLS equation with VBC based on HBDT
Soliton solution of the DNLS equation based on Hirota's bilinear derivative transformation. 193) which differs from the usual derivative, for example. The above equations contain the complete information needed to search for a soliton solution of the DNLS equation with VBC. In the end we get the one-soliton solution to the DNLS equation with VBC.
It is easy to find, up to an allowed constant global phase factor eiπ ¼ 1, the one-soliton solution (234) or (237) obtained in this paper is in perfect agreement with those obtained from other approaches [16]. By further redefining its soliton center, initial phase and λ1¼ρ1eiβ1, the one-soliton solution can be transformed into the usual typical form. Because what we are only concerned with are the soliton solutions, our soliton solution of DNLS compared to VBC is only a subset of the entire solution set.
Breather-type and pure N-soliton solution to DNLS + equation with NVBC based on revised IST
- The fundamental concepts for the IST theory of DNLS equation
Reference [7] built their theory by introducing a damping factor in the Zakharov-Shabat IST integral, to make it convergent, and further adopted a good idea of introducing an affine parameter to avoid the problem of the multi-valued problem in Riemann sheets, but both of their results are assumed in their independent soliton solutions and obtained from their soliton-phase7 and 9], so reference [8] had to verify an identity required by the form standard of a soliton solution (see expression (52) in Ref. [8]). On the other hand, the author of reference [8] also admits that his soliton solution does not have a rigorous verification of the standard form. Questions then naturally arise – whether the traditional IST for the DNLS equation with NVBC can be avoided and further improved.
The resulting one-soliton solution may naturally tend to the well-established conclusion of the VBC case asρ!0 [17–20] and the pure one-soliton solution in the degenerate case. Φðx,zÞ ¼Ψðx,zÞ Tð Þz (12) Some useful and important symmetry properties can be found. The relationship between the solution and the Jost functions of DNLS+equation The asymptotic behavior of the Jost solutions in the limit of j j !λ ∞can be.
- Introduction of time evolution factor
In order for the Jost functions to satisfy the second Lax equation, a time evolution factor h t,ð zÞ must be introduced via a standard procedure [21, 22] into the Jost functions and the scattering data. Nevertheless, hereafter the time variable is suppressed in Jost functions because it does not affect the treatment of the Z-S equation. In view of these failed experiences, we proposed a revised method to derive a suitable IST and the corresponding Z-S equation by multiplying an inverse spectral parameter 1=λ,λ¼ðzþρ2z 1Þ=2 before the Z-S integrand.
Using a standard procedure, time dependences of bn and c similar to (43) can be derived. 52) and (53) into formula (36) and let z!∞, we get the conjugate of the rawN-soliton solution (the time dependence is suppressed). Ið þBÞ 1: (65) Note that due to the choice of poles,zn, ðn¼1, 2,⋯,NÞ detðIþBÞ must be a nonzero and ðIþBÞinvertible matrix. On the other hand, Eq. 3) in the case of NVBC and without reflection (note that the time dependence of the soliton solution naturally appears incnð Þt.
Verification of standard form and the explicit breather-type multi-soliton solution
- The explicit N-soliton solution to the DNLS + equation with NVBC
As a result of (95), comparing (83) and (93) results in the expected identity and the verification of the first identity in (82) is completed. Aðn1,n2,⋯,nrÞ (116) where Aðn1,n2,⋯,nrÞ is the determinant of an r,th order minor ofAconsisting of elements belonging not only to rowsðn1,n2,⋯,nrÞ but also to columnsðn1,n2,⋯,nrÞ. By substituting (83) and (116) into (80), we finally achieve an explicit breather-type N-soliton solution of the DNLS + Eq. 1) with NVBC under reflectionless case, based on a revised and improved inverse scattering transformation.
Due to the limitation of space, the asymptotic behavior of the N-soliton solution is just similar to that of the pure N-soliton solution in Ref. 7] and therefore not discussed here, but it must be emphasized that in the limit of t. The one- and two-soliton solutions for DNLS+comparison with NVBCWe give two concrete examples – the one- and two-breath type soliton solutions.
The one and two-soliton solutions to DNLS + equation with NVBC We give two concrete examples – the one and two breather-type soliton solu-
- Explicit pure N-soliton solution to the DNLS + equation with NVBC
This confirms the validity of our N-Soliton solution formula and the reliability of the newly revised inverse scattering transform. The evolution of the two-soliton solution of the breathing series with respect to time and space is given in Fig. 3. This means that the N-soliton solution can be viewed as well-separated exact pure single solitons that line up with increasing order number, such as e.g.
The one- and two-soliton solutions are given as two typical examples to illustrate the unified formula of the N-soliton solution and the general calculation procedures. On the other hand, converging some/all of the poles in pairs on the circle in Figure 4 and rewriting the expression orð Þz as in can obviously generate the partially/fully pure multi-soliton solution. Finally, the elastic collision between the breathers of the above multi-soliton solution was demonstrated by the case of a breather-type 2-soliton solution.
Space periodic solutions and rogue wave solution of DNLS equation DNLS equation is one of the most important nonlinear integrable equations in
- Bilinear derivative transformation of DNLS equation
- Solution of bilinear equations
Space periodic solutions and rogue wave solution of DNLS equation DNLS equation is one of the most important nonlinear integrable equations in. For this we have completed the calculation of the 1st-order space periodic solution, the space-time evolution of its module is depicted in Figure 6. ffiffi ¼i,ρ¼ p2.
The space-time evolution of the module of the rogue wave solution with ρ¼1 and σ¼ ffiffiffi p2. The space-time evolution of the module of the 2nd order space periodic solution with. The space-time evolution of the module of the 2nd order space periodic solution (223) is shown in Figure 8. If we pay attention to the shape of this breather and the previous one, we will see that this breather can exactly degenerate into the 1st order breather if we take p3 ¼p1.
Concluding remarks
In this section, the first-order and second-order periodic spatial solutions of the KN equation are derived by means of HBDT. And after an integral transformation, these two breaths can be transferred to the solutions of the CLL equation. Meanwhile, based on the long-wave limit, the simplest rogue wave model is obtained according to the first-order periodic spatial solution.
Moreover, the generalized form of these two breaths is obtained, which gives us an instinctive speculation that higher-order periodic space solutions may hold this generalized form, but a precise demonstration is needed. Furthermore, higher order rogue wave models cannot be directly constructed from the long-wave limit of a higher order spatial periodic solution, because the higher order spatial periodic solution has multiple wavenumbers, we are also interested in searching for an alternative method besides DT that can help us determine higher order rogue wave solutions.
Appendices
- Atomic spin chain
- Atomic spin chain with magnetic dipole: dipole interaction
- Atomic spin chain with light-induced dipole-dipole interaction
- Magnetic soliton in atomic spin chain 1 Magnetic soliton under the MDDI
- Magnetic soliton under the LDDI
- Conclusion
The parameter Jmagmn describes the strength of the site-to-site spin coupling induced by the static MDDI. The realistic interaction strengths of the MDDI and LDDI coupling coefficients were reported in ref. From this equation, we can obtain the existence and dynamic development of the nonlinear magnetic soliton.
The spin coupling coefficients Jzmn¼Jmagmn and Jijas a function of the grating site index for different transverse widths of the condensate in the MDDI and LDDI dominating optical grating [4]. For two adjacent sites, A2 ¼Jmag02 =Jmag01 measures the relative strength of the NNN spin coupling to the NN spin coupling. Thus, for two adjacent sites, A2 ¼J02=J01 measures the relative strength of the NNN spin coupling to the NN spin coupling.
