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 Schr¨odinger equation.We show that this system is a completely integrable Hamiltonian system.
We use the 1-fold Darboux transformation (DT) of an inhomogeneous nonlinear Schro¨dinger equation (INLSE) to construct the deformed-soliton, breather, and rogue wave solutions explicitly. Furthermore, the obtained first-order deformed rogue wave solution, which is derived from the deformed breather solution through the Taylor expansion, is different from the known rogue wave solution of the nonlinear Schro¨dinger equation (NLSE). The effect of inhomogeneity is fully reflected in the variable height of the deformed soliton and the curved background of the deformed breather and rogue wave. By suitably adjusting the physical parameter, we show that a desired shape of the rogue wave can be generated. In particular, the newly constructed rogue wave can be reduced to the corresponding rogue wave of the nonlinear Schro¨dinger equation under a suitable parametric condition.
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 R4N|2N with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.
In this paper, considering the Hirota and the Maxwell-Bloch (H-MB) equations which are governed by femtosecond pulse propagation through a two-level doped fiber system, we construct the Darboux transformation of this system through a linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.
The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy(BTH).As a specific example of exponential solutions of the BTH,the authors consider a regular solution for the(1,2)-BTH with a 3×3-sized Lax matrix,and discuss some geometric structures of the solution from which the difference between the(1,2)-BTH and the original Toda hierarchy is shown.After this,the authors construct another kind of Lax representation of(N,1)-BTH which does not use the fractional operator of Lax operator.Then the authors introduce the lattice Miura transformation of(AT,1)-BTH which leads to equations depending on one field,and meanwhile some specific examples which contain the Volterra lattice equation(a useful ecological competition model)are given.
The solutions q [n] generated from a periodic "seed" q = cei(as+bt) of the nonlinear Schrdinger(NLS) by n-fold Darboux transformation is represented by determinant.Furthermore,the s-periodic solution and t-periodic solution are given explicitly by using q [1].The curves and surfaces(F1,F2,F3) associated with q [n] are given by means of Sym formula.Meanwhile,we show periodic and asymptotic properties of these curves.
