Multi-cut solutions of Laplacian growth

A new class of solutions to Laplacian growth (LG) with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut LG solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.     

Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles

The N-dimensional Hamiltonian

     

Additional Symmetries and String Equation of the CKP Hierarchy

Based on the Orlov and Shulman’s M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L ^l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra , which is a subalgebra of W _1+∞.      

Remarks on Exactly Solvable Noncommutative Quantum Field

We study exactly the solvable noncommutative scalar quantum field models of （2n） or （2n ＋ 1） dimensions. By writing out an equivalent action of the noncommutative field, it is shown that the special condition B. 0 = 4-1 in field theoretic context means the full restoration of the maximal U（∞） gauge symmetries broken due to kinetic term. It is further shown that the model can be obtained by dimensional reduction of a 2n-dimensional exactly solvable noncommutative φ4 quantum field model closely related to the 1＋1-dimensional Moyal/matrix-valued nonlinear Schr6dinger （MNLS） equation. The corresponding quantum fundamental commutation relation of the MNLS model is also given explicitly.     

On the BKP Hierarchy: Additional Symmetries,Fay Identity and Adler–Shiota–van Moerbeke Formula

We give an alternative proof of the Adler–Shiota–van Moerbeke formula for the BKP hierarchy. The proof is based on a simple expression for the generator of additional symmetries and the Fay identity of the BKP hierarchy.      

Non-isospectral integrable couplings of Ablowitz-Ladik hierarchy with self-consistent sources

A kind of new non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with is presented. As an application example, the integrable coupling hierarchy of non-isospectral Ablowitz-Ladik with self-consistent sources is derived by using of the loop algebra .     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.     

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.     

New block matrix spectral problem and Hamiltonian structure of the discrete integrable coupling system

In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity.     

He's homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients

In this Letter, He's homotopy perturbation method is applied to heat-like and wave-like equations with variable coefficients. The solutions are introduced in this Letter are in recursive sequence forms which can be used to obtain the closed form of the solutions if they are required. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method.