[Show abstract][Hide abstract] ABSTRACT: Considering certain terms of the next asymptotic order beyond the nonlinear
Schrodinger equation (NLS) equation, the Fokas-Lenells (FL) equation governed
by the FL system arise as a model for nonlinear pulse propagation in optical
fibers. The expressions of the q[n] and r[n] in the FL system are generated by
n-fold Darboux transformation (DT). Further, a Taylor series expansion about
the n-order breather solutions q[n] generated using DT by assuming periodic
seed solutions under reduction can generate the n-order rogue waves of the FL
equation explicitly with 2n+3 free parameters.
Mathematical Methods in the Applied Sciences 04/2015; 38(6). DOI:10.1002/mma.3133 · 0.92 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The mixed nonlinear Schrödinger (MNLS) equation is a model for the propagation of the Alfvén wave in plasmas and the ultrashort light pulse in optical fibers with two nonlinear effects of self-steepening and self phase-modulation(SPM), which is also the first non-trivial flow of the integrable Wadati-Konno-Ichikawa(WKI) system. The determinant representation Tn
of a n-fold Darboux transformation(DT) for the MNLS equation is presented. The smoothness of the solution q
[2k] generated by T
2k
is proved for the two cases (non-degeneration and double-degeneration ) through the iteration and determinant representation. Starting from a periodic seed(plane wave), rational solutions with two parameters a and b of the MNLS equation are constructed by the DT and the Taylor expansion. Two parameters denote the contributions of two nonlinear effects in solutions. We show an unusual result: for a given value of a, the increasing value of b can damage gradually the localization of the rational solution, by analytical forms and figures. A novel two-peak rational solution with variable height and a non-vanishing boundary is also obtained.
[Show abstract][Hide abstract] ABSTRACT: By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint, we obtain a finite dimensional system from the Lax pair of the nonlinear Schrödinger equation. We show that this system is a completely integrable Hamiltonian system.
[Show abstract][Hide abstract] ABSTRACT: The rotating reduced Maxwell-Bloch (RMB) equations, which describe the propagation of few-cycle optical pulses in a transparent media with two isotropic polarized electronic field components, are derived from a system of complete Maxwell-Bloch equations without using the slowly varying envelope approximations. Two hierarchies of the obtained rational solutions, including rogue waves, which are also called few-cycle optical rogue waves, of the rotating RMB equations are constructed explicitly through degenerate Darboux transformation. In addition to the above, the dynamical evolution of the first-, second-, and third-order few-cycle optical rogue waves are constructed with different patterns. For an electric field E in the three lower-order rogue waves, we find that rogue waves correspond to localized large amplitude oscillations of the polarized electric fields. Further a complementary relationship of two electric field components of rogue waves is discussed in terms of analytical formulas as well as numerical figures.
Physical Review E 12/2013; 88(6-1):062925. DOI:10.1103/PhysRevE.88.062925 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper, we construct the additional symmetries of one-component
constrained discrete KP (cdKP) hierarchy, and then prove that the algebraic
structure of the symmetry flows is the positive half of Virasoro algebra.
[Show abstract][Hide abstract] ABSTRACT: A new (1+1)-dimensional integrable system, i. e. the super coupled
Korteweg-de Vries (cKdV) system, has been constructed by a super extension of
the well-known (1+1)-dimensional cKdV system. For this new system, a novel
symmetry constraint between the potential and eigenfunction can be obtained by
means of the binary nonlinearization of its Lax pairs. The constraints for even
variables are explicit and the constraints for odd variables are implicit.
Under the symmetry constraint, the spacial part and the temporal parts of the
equations associated with the Lax pairs for the super cKdV system can be
decomposed into the super finite-dimensional integrable Hamiltonian systems on
the supersymmetry manifold $R^{4N|2N+2}$, whose integrals of motion are
explicitly given.
Journal of Physics A Mathematical and Theoretical 09/2010; 43(44). DOI:10.1088/1751-8113/43/44/445201 · 1.58 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP)
hierarchies are given by their actions on the Lax operators, and their actions
on the eigenfunction and adjoint eigenfunction $\{\Phi_i,\Psi_i \}$ are
presented explicitly. Furthermore, we show that acting on the space of the wave
operator, $\partial_k^*$ forms new centerless $W^{cC}_{1+\infty}$ and
$W^{cB}_{1+\infty}$-subalgebra of centerless $W_{1+\infty}$ respectively. In
order to define above symmetry flows $\partial_k^*$ of the cCKP and cBKP
hierarchies, two vital operators $Y_k$ are introduced to revise the additional
symmetry flows of the CKP and BKP hierarchies.
Science China Mathematics 07/2010; DOI:10.1007/s11425-010-4076-6 · 0.66 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out
for the super Dirac systems. Under the obtained symmetry constraint, the n-th flow of the super Dirac hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined
over the super-symmetry manifold R
4N|2N
with the corresponding dynamical variables x and t
n
. The integrals of motion required for Liouville integrability are explicitly given.
KeywordsSymmetry constraints-Binary nonlinearization-Super Dirac systems-Super finite-dimensional integrable Hamiltonian systems
2000 MR Subject Classification35Q51-37J35-37K10-37K40
Chinese Annals of Mathematics 05/2010; 31(3):361-372. DOI:10.1007/s11401-009-0032-6 · 0.45 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Based on the Adler-Shiota-van Moerbeke (ASvM) formula, the Virasoro
constraints and W-constraints for the p-reduced q-deformed
Kadomtsev-Petviashvili (q-KP) hierarchy are established.
[Show abstract][Hide abstract] ABSTRACT: This paper gives a recursion operator for a 1-constrained CKP hierarchy, and by the recursion operator it proves that the 1-constrained CKP hierarchy can be reduced to the mKdV hierarchy under condition $q=r$. Comment: To appear in "Acta Matematica Scientia"
Acta Mathematica Scientia 04/2010; 31(4). DOI:10.1016/S0252-9602(11)60316-0 · 0.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax equations,give the Sato equations, wave operators, Hirota bilinear identities (HBI) and show the existence of $tau$ function $\tau(t)$. Meanwhile we prove the validity of its Fay-like identities and Hirota bilinear equations (HBEs) in terms of vertex operators whose coefficients take values in the algebra of differential operators. In contrast with HBEs of the usual integrable system, the current HBEs are equations of product of operators involving $e^{\partial_x}$ and $\tau(t)$. Comment: 29 pages, to appear Journal of Mathematical Physics(2010)
[Show abstract][Hide abstract] ABSTRACT: Based on the Lax operator L and Orlov-Shulman’s M operator, the string equations of the q-KP hierarchy are established from the special additional symmetry flows, and the negative Virasoro constraint generators
{L
−n
, n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.
Keywords
q-KP hierarchy–Additional symmetry–String equations–Virasoro constraints
Chinese Annals of Mathematics 01/2009; 32(6):895-904. DOI:10.1007/s11401-011-0678-8 · 0.45 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A special chain of the gauge transformations of the two-component constrained KP is introduced, and then the transformed components and the τ-function of this hierarchy are presented. Based on this, by the different reductions, the gauge transformations of the constrained BKP and the constrained CKP hierarchies are obtained. Furthermore, the transformed component functions and τ-functions of them are expressed by the determinants with the help of the determinant representation of the gauge transformation operators.
[Show abstract][Hide abstract] ABSTRACT: We establish the binary nonlinearization approach of the spectral problem of the super AKNS system, and then use it to obtain the super finite-dimensional integrable Hamiltonian system in supersymmetry manifold $\mathbb{R}^{4N|2N}$. The super Hamiltonian forms and integrals of motion are given explicitly. Comment: 13pages, Latex, to appear in Modern Phys. Lett.B
Modern Physics Letters B 11/2007; 22(04). DOI:10.1142/S0217984908014778 · 0.75 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a systematic approach to the construction of soliton solutions for the 5-reduction of the C-type sub-hierarchy for the Kadomtsev–Petviashvili (CKP) hierarchies starting from the general τ-function τ(n+k) of the Kadomtsev–Petviashvili (KP) hierarchy. We obtain the one-soliton and two-soliton solutions for the bi-directional Kaup–Kupershmidt (bKK) equation, i.e. the 5-reduction of CKP hierarchy.
[Show abstract][Hide abstract] ABSTRACT: The n-fold Darboux transform (DT) is a 2×2 matrix for the Ablowitz-Kaup-Newell-Segur (AKNS) system. In this paper, each element
of this matrix is expressed by 2n + 1 ranks’ determinants. Using these formulae, the determinant expressions of eigenfunctions generated by the n-fold DT are obtained. Furthermore, we give out the explicit forms of the n-soliton surface of the Nonlinear Schrodinger Equation (NLS) by the determinant of eigenfunctions.
Science in China Series A Mathematics 12/2006; 49(12):1867-1878. DOI:10.1007/s11425-006-2025-1 · 0.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a systematic way to construct solutions of the (n=5)-reduction of the BKP and CKP hierarchies from the general tau function of the KP hierarchy. We obtain the one-soliton, two-soliton, and periodic solution for the bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK) and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions such as left- and right-going solitons are classified according to the symmetries of the 5th roots of exp(i epsilon). Furthermore, we show that the soliton solutions of the n-reduction of the BKP and CKP hierarchies with n= 2 j +1, j=1, 2, 3, ..., can propagate along j directions in the 1+1 space-time domain. Each such direction corresponds to one symmetric distribution of the nth roots of exp(i epsilon). Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian tau function of the sub-hierarchies BKP and CKP. If n is even, we again find two-peak solitons. Last, we obtain the "stationary" soliton for the higher-order KP hierarchy.
Journal of High Energy Physics 04/2005; DOI:10.1088/1126-6708/2006/03/103 · 6.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The determinant representation of the iterated Darboux transforms of the matrix AKNS hierarchy is established. For this purpose, the authors generalize the Sylvester identity, and use it to simplify the complex determinants which appear in the process of iterating Darboux transforms. Finally, the authors give the soliton solutions of some famous matrix soliton equations such as matrix KdV equation, matrix NLS equation and matrix KdV equation.