The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

We report discovery of collinear-magnetism-driven ferroelectricity in the Ising chain magnet Ca3Co2-xMn(x)O6 (x approximately 0.96). Neutron diffraction shows that Co2+ and Mn4+ ions alternating along the chains exhibit an up-up-down-down ( upward arrow upward arrow downward arrow downward arrow) magnetic order. The ferroelectricity results from the inversion symmetry breaking in the upward arrow upward arrow downward arrow downward arrow spin chain with an alternating charge order. Unlike in spiral magnetoelectrics where antisymmetric exchange coupling is active, the symmetry breaking in Ca3(Co,Mn)2O6 occurs through exchange striction associated with symmetric superexchange.

/pdf/ferroelectricity-in-an-ising-chain-magnet-1jpacpf3gz.pdf

Ferroelectricity in an Ising Chain Magnet

Water-wave symbolic computation for the Earth, Enceladus and Titan: The higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations

Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas

Optical waves/modes in a multicomponent inhomogeneous optical fiber via a three-coupled variable-coefficient nonlinear Schrödinger system

Under investigation is a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation, which describes the weakly dispersive waves in a fluid. Via the Hirota method and symbolic computation, we obtain the mixed lump-kink and mixed rogue wave-kink solutions. Through the mixed lump-kink solutions, we observe three different phenomena between a lump and one kink. For the fusion phenomenon, a lump and a kink are merged with the lump’s energy transferring into the kink gradually, until the lump merges into the kink completely. Fission phenomenon displays that a lump separates from a kink. The last phenomenon shows that a lump travels together with a kink with their amplitudes unchanged. In addition, we graphically study the interaction between a rogue wave and a pair of the kinks. It can be observed that the rogue wave arises from one kink and disappears into the other kink. At certain time, the amplitude of the rogue wave reaches the maximum.

Mixed lump-kink and rogue wave-kink solutions for a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics

In this paper, investigation is made on the coupled variable-coefficient fourth-order nonlinear Schrodinger equations, which describe the simultaneous propagation of optical pulses in an inhomogeneous optical fiber. Via the generalized Darboux transformation, the first- and second-order rogue wave solutions are constructed. Based on such solutions, effects of the group velocity dispersion coefficient and the fourth-order dispersion coefficient on the rogue waves are graphically analyzed. The first-order rogue waves with the eye-shaped distribution, the interactions between the first-order rogue waves with solitons, and the second-order rogue waves with one highest peak and with the triangular structure are displayed. When the value of the group velocity dispersion coefficient or the fourth-order dispersion increases, range of the first-order rogue wave increases. Composite rogue waves are obtained, where the temporal separation between two adjacent rogue waves can be changed if we adjust the group velocity dispersion coefficient and fourth-order dispersion coefficient. Periodic rogue waves are presented. Periods of such rogue waves decrease with the periods of the group-velocity dispersion and fourth-order dispersion coefficient decreasing.

Rogue waves for the coupled variable-coefficient fourth-order nonlinear Schrödinger equations in an inhomogeneous optical fiber

Under investigation in this paper is a fifth-order defocusing nonlinear Schrodinger equation for the attosecond pulses in the optical fiber communication. Lax pair, one/N-fold ( N = 2 , 3 , … ) bin...

Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication

Dark–bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrödinger system in an optical fiber

In this paper, we investigate the coupled Sasa-Satsuma equations, which describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Darboux-dressing transformation is applied to obtain the dark-bright soliton and semirational rogue-wave solutions. Dark-bright one solitons with the single-hump, double-hump, and even breather-like structures are presented. Interactions between the double-peak breather and different kinds of dark-bright solitons are studied. We show that the double-peak (or single-peak) rogue wave can coexist and interact with different kinds of dark-bright solitons. Coexistence of the solitons with different velocities and rogue waves is also found. Numerical stabilities of the dark-bright solitons and semirational rogue waves are exhibited. It is expected that those localized wave phenomena can be experimentally observed and have potential applications.

Zhong Du

Papers

Mixed lump-kink and rogue wave-kink solutions for a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics

Rogue waves for the coupled variable-coefficient fourth-order nonlinear Schrödinger equations in an inhomogeneous optical fiber

Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrödinger equation for the attosecond pulses in the optical fiber communication

Dark–bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrödinger system in an optical fiber

Dark-bright solitons and semirational rogue waves for the coupled Sasa-Satsuma equations.