1-Soliton solution of the generalized KP equation with generalized evolution

This paper obtains the 1-soliton solution of three variants of the generalized KP equation with generalized evolution. The solitary wave ansatz is used to carry out the integration of such equations. The parameter domain is also identified in the process. The numerical simulations are also obtained in this paper.     

1-Soliton solution of the generalized Burgers equation with generalized evolution

The generalized Burgers equation with generalized evolution is studied in this paper. The soliton ansatz is used to carry out the integration of this generalized Burgers equation. This study is then extended to 1+2 dimensions, even with full nonlinearity.     

Metric regularity of a positive order for generalized equations

This paper focuses on the metric regularity of a positive order for generalized equations. More concretely, we establish verifiable sufficient conditions for a generalized equation to achieve the metric regularity of a positive order at its a given solution. The provided conditions are expressed in terms of the Fréchet coderivative/or the Mordukhovich coderivative/or the Clarke one of the corresponding multifunction formulated the generalized equation. In addition, we show that such sufficient conditions turn out to be also necessary for the metric regularity of a positive order of the generalized equation in the case where the multifunction established the generalized equation is closed and convex.     

Solitary wave solution for the generalized Kawahara equation

The travelling wave ansatz is used to find the solitary wave solution of the generalized Kawahara equation. The ansatz is obtained from the structure of the soliton solution of the Kawahara equation and the modified Kawahara equation. The first two integrals of motion of the generalized Kawahara equation are also computed in this work.     

Hamiltonian structures of the generalized cylindrical KdV equations

Hamiltonian structures of the cylindrical Korteweg-deVries equation and its higher order equations are found. The connection between the generalized C-KdV equation and nonisospectral problem is pointed out.Projects supported by the Science Fund of the Chinese Academy of Sciences.     

Group properties of generalized quasi-linear wave equations

In this paper, complete group classification of a class of (1+1)-dimensional generalized quasi-linear wave equations is performed by using the Lie-Ovsiannikov method, additional equivalent transformation and furcate split method. Lie reductions of some truly ‘variable coefficient’ wave equations which are singled out from the classification results are investigated. Some classes of exact solutions of these ‘variable coefficient’ wave equations are constructed by means of both the reductions and the additional equivalent transformations. The nonclassical symmetries to the generalized quasi-linear wave equation are also studied. This enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries.     

Numerical treatment of the spherically symmetric solutions of a generalized Fisher-Kolmogorov-Petrovsky-Piscounov equation

In the present work, the connection of the generalized Fisher-KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher-KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher-KPP equation, the Heaviside equation, and the generalized Fisher-KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher-KPP problems.     

A generalized Fitzhugh–Nagumo equation

In this paper we investigate mappings of the classical Fitzhugh–Nagumo equation to a generalized Fitzhugh–Nagumo equation. These mappings are invertible and transform the solutions of the classical Fitzhugh–Nagumo equation into solutions of the generalized Fitzhugh–Nagumo equation considered here. These mappings are found by considering the Lie point symmetries admitted by the classical Fitzhugh–Nagumo equation and the generalized Fitzhugh–Nagumo equation considered here. A particular example of a generalized Fitzhugh–Nagumo equation that satisfies the boundary conditions of the classical Fitzhugh–Nagumo equation is considered. Numerical solutions of the generalized Fitzhugh–Nagumo equation that do not satisfy the boundary conditions of the classical Fitzhugh–Nagumo equation are obtained by implementing the Method of Lines.     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.     

Some exact Wick type stochastic generalized Boussinesq equation solutions

The resolution of the stochastic generalized Boussinesq equation driven by a white noise is undertaken. Explicit solutions are found thanks to a white noise functional approach and the F-expansion method. Among these solutions, periodic and solitonic ones are pointed out.     

Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation

The bounded traveling wave solutions of a generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with $p=2$ and $p=3$ both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.     

A generalized AKNS hierarchy,bi-Hamiltonian structure,and Darboux transformation

A new generalized AKNS hierarchy is presented starting from a 4 × 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given.     

A parallel algorithm for generalized networks

This paper presents an application of parallel computing techniques to the solution of an important class of planning problems known as generalized networks. Three parallel primal simplex variants for solving generalized network problems are presented. Data structures used in a sequential generalized network code are briefly discussed and their extension to a parallel implementation of one of the primal simplex variants is given. Computational testing of the sequential and parallel codes, both written in Fortran, was done on the CRYSTAL multicomputer at the University of Wisconsin, and the computational results are presented. Maximum efficiency occurred for multiperiod generalized network problems where a speedup approximately linear in the number of processors was achieved.This research was supported in part by NSF grants DCR-8503148 and CCR-8709952 and by AFOSR grant AFOSR-86-0194.     

Implicit solitary wave solutions of the generalized magma equation

The generalized magma equation, in which dispersion and nonlinearity are coupled together in a manner reminiscent of the Harry-Dym equation, is solved to indicate both periodic and aperiodic implicit solitary wave solutions. It is argued that the presence of an additional spatial derivative in the magma equation also permits explicit solitary wave solutions, a feature not shared by the Harry-Dym equation.     

Discrete equation on a square lattice with a nonstandard structure of generalized symmetries

We clarify the integrability nature of a recently found discrete equation on the square lattice with a nonstandard symmetry structure. We find its L-A pair and show that it is also nonstandard. For this discrete equation, we construct the hierarchies of both generalized symmetries and conservation laws. This equation yields two integrable systems of hyperbolic type. The hierarchies of generalized symmetries and conservation laws are also nonstandard compared with known equations in this class.     

On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation are successfully obtained, which contain soliton-like and periodic-like solutions This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

New variants of the generalized level method for minimization of convex nondifferentiable functions taking infinite values

New variants of the generalized level method for minimization of convex Lipschitz functions on a compact set with a nonempty interior are proposed. These variants include the well-known generalized and classical level methods. For the new variants, an estimate of the convergence rate is found, including the variants in which the auxiliary problems are solved approximately.     

Topological 1-soliton solution of the generalized KdV equation with generalized evolution

In this paper, the topological 1-soliton solution to the generalized Korteweg-de Vries equation with generalized evolution will be obtained. The solitary wave ansatz methods will be applied to obtain the solutions. In the process, the constraints relations between the soliton parameters will also be determined.     

Bright and dark solitons of the generalized nonlinear Schrödinger’s equation

This paper studies the generalized form of the nonlinear Schrödinger’s equation. The special cases of Kerr law, power law, parabolic law and the dual-power laws are considered. The 1-soliton solution is obtained in all of these four cases. The adiabatic parameter dynamics of the solitons due to perturbation terms are laid down. In addition, the analysis of dark soliton is also carried out. Finally, a few numerical simulations of these equations are given.     

Exact solitary wave solutions for a generalized KdV–Burgers equation

Using the special truncated expansion method, the solitary wave solutions are constructed for the compound Korteweg–de Vries–Burgers (KdVB) equation. Exact and explicit solitary wave solutions for a generalized KdVB equation are obtained by introducing a suitable ansatz equation. The generalized two-dimensional KdVB equation is discussed. Some particular cases of the generalized KdVB equation are solved by using these methods.