Soliton Interactions of the Kadomtsev–Petviashvili Equation and Generation of Large-Amplitude Water Waves

We study the maximum wave amplitude produced by line-soliton interactions of the Kadomtsev–Petviashvili II (KPII) equation, and we discuss a mechanism of generation of large amplitude shallow water waves by multi-soliton interactions of KPII. We also describe a method to predict the possible maximum wave amplitude from asymptotic data. Finally, we report on numerical simulations of multi-soliton complexes of the KPII equation which verify the robustness of all types of soliton interactions and web-like structure.     

Kadomtsev–Petviashvili Equation Revisited and Integrability in 4 + 2 and 3 + 1

The Cauchy problem of Kadomtsev–Petviashvili I (KPI) was reduced to a nonlocal Riemann–Hilbert (RH) problem by the author and Ablowitz in 1983. This formulation was based on the introduction of two spectral functions (nonlinear Fourier transforms, FTs). This formalism was improved by Boiti et al. [ 1 ], where it was shown that the earlier nonlocal RH problem can be formulated in terms of a single spectral function (nonlinear FT). A different formalism was presented by Zhou [ 2 ], where the Cauchy problem was rigorously solved in terms of a linear integral equation involving a nonanalytic eigenfunction. Here, we first revisit the above results and then review some recent results about the derivation of integrable generalizations of KP in 4 + 2 (i.e., in four spatial and two temporal dimensions), as well as in 3 + 1 (i.e., in three spatial and one temporal dimensions).     

Nonexistence Results for the Korteweg–de Vries and Kadomtsev–Petviashvili Equations

We study characteristic Cauchy problems for the Korteweg–de Vries (KdV) equation u_t = uu_x + u_xxx , and the Kadomtsev–Petviashvili (KP) equation u_yy =( u_xxx + uu_x + u_t ) _x with holomorphic initial data possessing non-negative Taylor coefficients around the origin. For the KdV equation with initial value u (0,  x )= u ₀( x ), we show that there is no solution holomorphic in any neighborhood of ( t ,  x )=(0, 0) in C² unless u ₀( x )= a ₀+ a ₁ x . This also furnishes a nonexistence result for a class of y -independent solutions of the KP equation. We extend this to y -dependent cases by considering initial values given at y =0, u ( t ,  x , 0)= u ₀( x ,  t ), u_y ( t ,  x , 0)= u ₁( x ,  t ), where the Taylor coefficients of u ₀ and u ₁ around t =0, x =0 are assumed non-negative. We prove that there is no holomorphic solution around the origin in C³, unless u ₀ and u ₁ are polynomials of degree 2 or lower. MSC 2000: 35Q53, 35B30, 35C10.     

Kadomtsev–Petviashvili Equation on the Half-Plane

We study the boundary value problem for the Kadomtsev–Petviashvili equation on the half-plane y > 0 with a homogeneous condition along the boundary. We show that the problem can be efficiently solved using the dressing method. We present explicit solutions for particular cases of the boundary value problem.     

The Cauchy Problem for the Kadomtsev–Petviashili II Equation with Nondecaying Data along a Line

The initial value problem for the Kadomstev–Petviashvili II (KPII) equation is considered with given data that are nondecaying along a line. The associated direct and inverse scattering of the two-dimensional heat equation is constructed. The direct problem is formulated in terms of a bounded Green's function. The inverse data are decomposed into scattering data along the line and     data from the decaying portion of the potential. The solution of the KPII equation is then obtained via coupled linear integral equations.     

Some Closed-Form Solutions of Burgers’ Equation

This paper shows how to use the method of quasisolutions to construct exact solutions to Burgers’ equation. A function υ=υ(x, y) is called a quasisolution of a PDE in case there exists a function φ (not a constant function) of one variable so that u(x, y)=φ(υ(x, y)) is a solution of the equation. We prove a theorem giving necessary and sufficient conditions for υ to be a quasisolution to Burgers’ equation. A function φ can then be found explicitly so that u=φ(υ) is an actual solution. Combining this technique with similarity methods, we find a continuum of solutions to Burgers' equation.     

Initial Time Layers and Kadomtsev–Petviashvili-Type Equations

The Kadomtsev–Petviashvili (KP) equation and generalizations (GKP) have temporal discontinuities at the initial instant of time. Motivated by the study of water waves, a generalized Boussinesq equation that contains the GKP equations as an "outer" limit is introduced. Within the context of matched asymptotic expansions the discontinuities are resolved. The linear system is analyzed in more detail and the limit process is rigorously established.     

Kac方程的解及其性质

研究Kac方程的初值问题．证明了该类方程存在唯一的全局分布解．并且使用一种新的线性化方法证明了该类方程的解具有相应的多项式衰减性．     

Long-time Solutions of the Ostrovsky Equation

The Ostrovsky equation is a modification of the Korteweg-de Vries equation which takes account of the effects of background rotation. It is well known that the usual Korteweg-de Vries solitary wave decays and is replaced by radiating inertia gravity waves. Here we show through numerical simulations that after a long-time a localized wave packet emerges as a persistent and dominant feature. The wavenumber of the carrier wave is associated with that critical wavenumber where the underlying group velocity is a minimum (in absolute value). Based on this feature, we construct a weakly nonlinear theory leading to a higher-order nonlinear Schrödinger equations in an attempt to describe the numerically found wave packets.     

Perturbation Solutions of the Whittaker-Hill Equation

When the Helmholtz equation ²V+k²V = 0 is separated in the generalparaboloidal co-ordinate system, the three ordinary differentialequations obtained each take, after a suitable change of variable,the form of the Whittaker-Hill equation. For the case k²<0,a considerable amount is known about the periodic solutionsof this equation. For k²>0, however, very little is so farknown. In this paper solutions of the Whittaker-Hill equationfor small positive k² are derived. These are the first explicitsolutions to be obtained for the case k²>0, and they couldbe employed to solve the Dirichlet or Neumann problem for ageneral paraboloid when k² is small. Three limiting cases arenoted, involving the reduction of the solutions to Mathieu functionsand the reduction of the co-ordinate system to the rotation-paraboloidalsystem.     

Asymptotic Solutions of the Korteweg-deVries Equation

The long-time asymptotic solution of the Korteweg-deVries equation, corresponding to initial data which decay rapidly as |x|→∞ and produce no solitons, is found to be considerably more complicated than previously reported. In general, the asymptotic solution consists of exponential decay, similarity, rapid oscillations and a “collisionless shock” layer. The wave amplitude in this layer decays as [(lnt)/t]^2/3. Only for very special initial conditions is the shock layer absent from the solution.     

Stationary Solutions of the KdV Equation

In this article, as a first step to study the large time behavior of solutions to the KdV equation on a half-line, we prove the existence of the stationary solutions to the corresponding problem. The aim is to concentrate on understanding the influence of boundary conditions.     

有限域上Carlitz方程的解数

设F_q是含有q个元素的有限域,其中q=p~t,t≥1,p是一个奇素数.研究了Carlitz方程的推广形式(a_1x_1~(m_1)+…+a_nx_n~(m_n)+a_(n+1)x_(n+1)~(m_(n+1))+…+a_(n+s)x_(n+s)~(m_(n+s)))~k=bx_1~(k_1)…x_n~(k_n),其中ai,b∈F_q~*,s≥1,n≥1.当方程变量的指数满足一定条件时,得到了方程的解数公式.     

The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation

A detailed analysis is given to the solution of the cubic Schrödinger equation iq_t + q_xx + 2|q|²q = 0 under the boundary conditions as |x|→∞. The inverse-scattering technique is used, and the asymptotic state is a series of solitons. However, there is no soliton whose amplitude is stationary in time. Each soliton has a definite velocity and “pulsates” in time with a definite period. The interaction of two solitons is considered, and a possible extension to the perturbed periodic wave [q(x + T,t) = q(x,t) as |x|→∞] is discussed.     

RLW—Burgers方程的精确解

借助未知函数的变换，ＲＬＷ－Ｂｕｒｇｅｒｓ方程和ＫｄＶ－Ｂｕｒｇｅｒｓ方程化为易于求解的齐次形式的方程，从而得到ＲＬＷ－Ｂｕｒｇｅｒｓ方程和ＫｄＶ－Ｂｕｒｇｅｒｓ方程的精确解。     

Lin- Reissner-Tsien方程的自相似解

Lin-Reissner-Tsien方程描述了在跨音速近似下的不稳定跨音速流动.通过相似变换,将Lin-Reissner-Tsien方程简化为一个常微分方程.然后解析求解所得到的方程,并在某些情况下得到的正好是精确解.上述情况下无法得到精确解时,给出了数值模拟.还得到了行波解.