다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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)

This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.

We study standard and nonlocal nonlinear Schrodinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions, respectively. By using the Hirota bilinear method, we first find soliton solutions of the coupled NLS system of equations; then using the reduction formulas, we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)|2 for the standard and nonlocal NLS equations.

Nonlocal nonlinear Schrödinger equations and their soliton solutions

General soliton solutions to a nonlocal nonlinear Schrodinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions are considered via the combination of Hirota's bilinear method and the Kadomtsev–Petviashvili (KP) hierarchy reduction method. First, general N-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.

https://iopscience.iop.org/article/10.1088/1361-6544/aae031/pdf

General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

In this paper, we prove that an integrable nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity 29, 915–946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one can see that there exists significant difference between the nonlocal complex mKdV equation and the classical complex mKdV equation. Through constructing the Darboux transformation for nonlocal complex mKdV equation, a variety of exact solutions including dark soliton, W-type soliton, M-type soliton, and periodic solutions are derived.

Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries equation

In the present paper, we study the real and complex coupled dispersionless (CD) equations, the real and complex short pulse (SP) equations geometrically and algebraically. From the geometric point of view, we first establish the link of the motions of space curves to the real and complex CD equations, then to the real and complex SP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. In the second part of the paper, it is made clear for the connection between the real and complex CD and SP equations and the two-component extended Kadomtsew-Petviashvili (KP) hierarchy. As a by-product, the N-soliton solutions in the form of determinants for these equations are provided.

From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

Lie symmetry analysis of the time fractional KdV-type equation

In this paper, the dimension of invariant subspaces admitted by nonlinear systems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator $\mathbb{F} = (F^1 ,F^2 )$
 with orders {k
 1, k
 2} (k
 1 ≥ k
 2) preserves the invariant subspace $W_{n_1 }^1 \times W_{n_2 }^2 (n_1 \geqslant n_2 )$
 , then n
 1 − n
 2 ≤ k
 2, n
 1 ≤ 2(k
 1 + k
 2) + 1, where $W_{n_q }^q $
 is the space generated by solutions of a linear ordinary differential equation of order n
 q
 (q = 1, 2). Several examples including the (1+1)-dimensional diffusion system and Ito’s type, Drinfel’d-Sokolov-Wilson’s type and Whitham-Broer-Kaup’s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.

Maximal dimension of invariant subspaces to systems of nonlinear evolution equations

In this paper, we prove that the nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity, 29, 915-946 (2016)] is gauge equivalent to a spin-like model From the gauge equivalence, one can see that there exists significant difference between the nonlocal complex mKdV equation and the classical complex mKdV equation Through constructing the Darboux transformation(DT) for nonlocal complex mKdV equation, a variety of exact solutions including dark soliton, W-type soliton, M-type soliton and periodic solutions are derived

Shoufeng Shen

Papers

Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries equation

From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

Lie symmetry analysis of the time fractional KdV-type equation

Maximal dimension of invariant subspaces to systems of nonlinear evolution equations

Integrable nonlocal complex mKdV equation: soliton solution and gauge equivalence