For the Kadomtsev–Petviashvili (KP) hierarchy constructed in terms of the famous Sato theory, a ‘‘k constraint’’ is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates. The eigenvalue problem of the k-constrained system is naturally obtained from the linear system of the KP hierarchy, which takes the form of kth-order polynomial coupled with a first-order one, thus we are able to derive the correspondent Lax pair, recursion operator, bi-Hamiltonian structures, and conserved quantities. The constraints for the BKP hierarchy are also sketched.

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We propose a new exactly solvable potential which consists of the modified Kratzer potential plus a new ring-shaped potential βctg 2 θ/r 2 . The exact solutions of the bound states of the Schrödinger equation for this potential are presented analytically by using the Nikiforov-Uvarov method, which is based on solving the second-order linear differential equation by reducing to a generalized equation of hypergeometric type. The wavefunctions of the radial and angular parts are taken on the form of the generalized Laguerre polynomials and the total energy of the system is different from the modified Kratzer potential because of the contribution of the angular part.

The n-fold Darboux transform (DT) is a 2×2 matrix for the Ablowitz-Kaup-NewellSegur (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.

In this paper, the partially party-time (PT ) symmetric nonlocal Davey-Stewartson (DS) equations with respect to x is called x-nonlocal DS equations, while a fully PT symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x-nonlocal DS equations, the usual (2 + 1)-dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)-dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.

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Liquid-phase-exfoliation technology was utilized to prepare layered MoS 2 , WS 2 , and MoSe 2 nanosheets in cyclohexylpyrrolidone. The nonlinear optical response of these nanosheets in dispersions was investigated by observing spatial self-phase modulation (SSPM) using a 488 nm continuous wave laser beam. The diffraction ring patterns of SSPM were found to be distorted along the vertical direction right after the laser traversing the nanosheet dispersions. The nonlinear refractive index of the three transition metal dichalcogenides dispersions n 2 was measured to be ∼10 −7 cm 2 W −1 , and the third-order nonlinear susceptibility χ 3 ∼ 10 −9 esu. The relative change of effective nonlinear refractive index Δn 2e ∕n 2e of the MoS 2 , WS 2 , and MoSe 2 dispersions can be modulated 0.012-0.240, 0.029-0.154, and 0.091-0.304, respectively, by changing the incident intensities. Our experimental results imply novel potential application of two-dimensional transition metal dichalcogenides in nonlinear phase modulation devices.

Spatial self-phase modulation (SSPM) was observed directly when a focused He-Ne laser beam at 633 nm went through liquid-phase-exfoliated graphene dispersions. The diffraction pattern of SSPM was found to be distorted rapidly right after the incident beam horizontally passing through the dispersions, while no distortion for the vertically incident geometry. We show that the distortion is originated mainly from the non-axis-symmetrical thermal convections of the graphene nanosheets induced by laser heating, and the relative change of nonlinear refractive index can be determined by the ratio of the distortion angle to the half-cone angle. Therefore, the effective nonlinear refractive index of graphene dispersions can be tuned by changing the incident intensity and the temperature of the dispersions.