Modified Korteweg-De Vries equation for spatially inhomogeneous plasmas

The problem of propagation of ion-acoustic KdV solitons in weakly, spatially inhomogeneous plasmas is considered taking into account self-consistently the zeroth-order velocity and the potential existing in the system due to the presence of the inhomogeneity. An explicit soliton solution of the modified KdV equation is obtained.     

Asymptotic behavior of the solutions of the Kadomtsev-Pyatviashvili equation (two-dimensional Korteweg-De Vries equation)

The asymptotic form of the solutions of the Kadomtsev-Pyatviashvili equation for t → ± ∞ is presented. The reverse problem of reconstructing the solution from its asymptotic form is also solved.     

On the inverse scattering transform,the cylindrical Korteweg-De Vries equation and similarity solutions

It is shown that the inverse scattering transform for the cylindrical KdV equation can be obtained directly from that for the 2-D KdV equation. Using this transform for the similarity solution, we obtain two representations of a Painlevé transcendent.     

A singular perturbation theory for the FitzHugh-Nagumo nerve conduction equation

A singular perturbation theory is applied to the FitzHugh-Nagumo nerve conduction equation with cubic nonlinearity to obtain jacobian elliptic-function travelling-wave profiles. Starting with zeroth-order elliptic-function solutions, we obtain explicit first-order correction to the propagating waves and pulses.     

Stability for the Korteweg-de Vries equation by inverse scattering theory

The stability problems for the Korteweg-de Vries equation, where linear stability fails, are investigated by the inverse scattering method. A rather general solution u(t, x) of the K-dV equation is shown to depend, for fixed time t, continuously on the initial condition u(0, x). For a continuum solution u_c(t, x), this continuity holds uniformly in t (stability), but for a soliton solution this is not true. A soliton solution can be uniquely decomposed into a continuum and discrete (soliton) part: u(t, x) = u_e(t, x) + u_d(t, x). Then the perturbed solution u is close to u after a suitable t-dependent “shift” of the soliton part (form stability).     

Landau-Ginzburg-Higgs方程的微扰理论

求出了一阶和二阶近似下微扰对Landau-Ginzburg-Higgs方程孤子解的影响，即求出了孤子参数随时间的缓慢变化关系及解的一阶修正和二阶修正的具体表达式. 关键词： Landau-Ginzburg-Higgs方程 孤子 微扰     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Conservation laws for the Korteweg-de Vries equation and the theory of partitions

We show how the terms appearing in the expressions for the densities and the fluxes for the Korteweg-de Vries equation may be found by combinatorial methods. Our basic device consists in associating partitions and their Ferrers graphs to the first density and to the first flux, and then in proceeding inductively following very simple rules. Furthermore, we use unrestricted partitions and a recurrence relation to specify every term of every integral power of the Sturm-Liouville (or one-dimensional Schrödinger) operator.     

Korteweg-De Vries Equation for Non-Linear Ion-Acoustic Waves in a Plasma with Negative Ions

The theoretical analysis, based on the perturbation technique, of ion-acoustic waves in the vicinity of a Korteweg-de Vries (K-dV) equation derived in a plasma with some negative ions has been made. The investigation shows that the negative ions in plasma with isothermal electrons introduced a critical concentration at which the ion-acoustic wave plays an important role of wave-breaking and forming a precursor while the plasma with non-isothermal electrons has no such singular behaviour of the wave. These two distinct features of ion waves lead to an overall different approach of present study of ion-waves. A distinct feature of non-uniform transition from the nonisothermal case to isothermal case has been shown. Few particular plasma models have been chosen to show the characteristics behaviour of the ion-waves existing in different cases.     

Variational perturbation theory for Fokker-Planck equation with nonlinear drift

We develop a recursive method for perturbative solutions of the Fokker-Planck equation with nonlinear drift. The series expansion of the time-dependent probability density in terms of powers of the coupling constant is obtained by solving a set of first-order linear ordinary differential equations. Resumming the series in the spirit of variational perturbation theory we are able to determine the probability density for all values of the coupling constant. Comparison with numerical results shows exponential convergence with increasing order.     

Breather-like solution of the Korteweg-de Vries equation

A new real singular solution of the Korteweg-de Vries equation is briefly described. It is shown that the spectrum of the associated linear problem consists of a pair of complex conjugate eigenvalues. The existence of constants of motion and eigenfunction normalization integral is also discussed.     

A new perturbation theory for transport coefficients

The Green's function method for the calculation of orbital energies developed in the preceding paper is applied to the benzene molecule. It is confirmed that the lowest-order approximation for irreducible diagrams in our theory is equivalent to the usual self-consistent field theory. Higherorder corrections to orbital energies are calculated and theoretical and experimental results are compared.     

Local field equation forA 4-coupling in renormalized perturbation theory

For the model ofA ⁴-coupling a finite form of the local field equation is proposed and checked in renormalized perturbation theory.The research reported in this paper was supported in part by the National Science Foundation.     

Bäcklund transformations for the cylindrical Korteweg-de Vries equation

In this letter we derive a Bäcklund transformation (BT) for the cylindrical KdV equation, show that this equation is the only one of a certain class possessing a BT and derive a second BT from the first. Finally we investigate the existence of BTs for generalized MKdV equations.     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

Dynamical symmetries and conservation laws for Korteweg-de Vries equation

The KdV-equation in two space time dimensions with the set of rapidly decreasing test functions as initial conditions is treated in the setting of nonlinear group and Lie algebra representations. The topological properties of the direct and inverse scattering mappings are discussed in detail.The algebra of continuous constants of motion turns out to be generated as in the linear case by three constants of motion and an extension of a representation of the e₂ Lie algebra on space-time symmetries to its enveloping algebra. The integrability of these representations is studied.It is further proved that the “moment problem” does not have a unique solution in this setting.The existence of noncommutative algebras of smooth time independent constants of motion is pointed out.     

Positons of the modified Korteweg de Vries equation

The concept of positons, i.e. certain multiparametric solutions of the Korteweg de Vries equation with new properties, is extended to the modified Korteweg de Vries equation. It is shown that the essential features of positons carry over to this case; the collision of positons, the solitary-wave-positon interaction and simple generalizations are discussed in detail. Suggestions for future research and possible applications of the present work are sketched.     

Hartree-Fock perturbation theory for systems with one-particle perturbation

The Hartree-Fock perturbation theory for theN-electron system with a one-particle perturbation is rederived using the resolvent operator formalism. It is shown that the second-order contribution to the total energy can be expressed in a compact form using a properly defined effective one-particle operator. Relations of the Hartree-Fock perturbation theory with both the many-body theory and the regular Hartree-Fock formalism are discussed.