Exact multi-wave solutions for the KdV equation

By means of the known Darboux transformation, starting from some stationary wave solutions, we construct a large number of new explicit multiple waves solutions to the Korteweg–de Vries equation, including stationary periodic–soliton solutions, stationary soliton–periodic solutions, doubly periodic solutions, triply periodic solutions, as well as two- and three-soliton solutions.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

New no-traveling wave solutions for the Liouville equation by Bäcklund transformation method

With the aid of the known Bäcklund transformation, starting from some given traveling solutions, we consider new exact no-traveling wave solutions to the Liouville equation, and a series of breather soliton solutions, doubly periodic solutions, two-soliton solutions as well as periodic-soliton solutions are obtained.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR GENERALIZED DRINFELD-SOKOLOV EQUATIONS

Ansatz method and the theory of dynamical systems are used to study the traveling wave solutions for the generalized Drinfeld-Sokolov equations. Under two groups of the parametric conditions, more solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions are obtained. Exact explicit parametric representations of these travelling wave solutions are given.     

Exact and soliton solutions to nonlinear transmission line model

A nonlinear transmission line (NLTL) is comprised of a transmission line periodically loaded with varactors, where the capacitance nonlinearity arises from the variable depletion layer width, which depends both on the DC and AC voltages of the propagating wave. An equivalent circuit model of NLTL is discussed analytically, in this article, and different type of solutions are celebrated. The improved extended tanh-function method has been applied successfully to extract the solutions. The obtained solutions are solitary wave solutions, singular periodic solutions, singular soliton solutions, Jacobi elliptic doubly periodic type solutions and Weierstrass elliptic doubly periodic type solutions. It is a very convenient tool to study the propagation of electrical solitons which propagate in the form of voltage waves in nonlinear dispersive media.     

Explicit solutions to the coupled KdV equations with variable coefficients

1　IntroductionandtheProblemPresentedSeekingtheexplicitsolutionofthenonlinearpartialdifferentialequation (NPDEs)isanimportantsubjectinsolitontheoryanditsapplication .Formanyyears,themainattentionwaspaidtotheconstantcoefficientNPDEs[1～ 7],manypowerfulmethodshavebeenproposedanddeveloped.Inrecentyears,moreandmoreattentionshavebeenpaidtovariablecoefficientNPDEs[8～ 13].Manymethodssuchassimilarityreductionmethod ,truncatedexpansionmethodandhomogeneousbalancemethodhavebeenextendedtosolvevaria…     

Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.     

Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

We prove that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that classical solutions of the latter exist. The method is based on a new matched asymptotic expansion for solutions, a spectral analysis for linearizd operators, and an estimate for the difference between the true solutions and certain approximate ones.     

Analytical study of solitons to nonlinear time fractional parabolic equations

The some of the well-known nonlinear time fractional parabolic partial differential equations is studied in this paper. The fractional complex transform and the first integral method are employed to construct one-soliton solutions of these equations. The power of this manageable method is confirmed. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions.     

A generalized auxiliary equation method and its applications

In this paper, a generalized auxiliary equation method with the aid of the computer symbolic computation system Maple is proposed to construct more exact solutions of nonlinear evolution equations, namely, the higher-order nonlinear Schrödinger equation, the Whitham–Broer–Kaup system, and the generalized Zakharov equations. As a result, some new types of exact travelling wave solutions are obtained, including soliton-like solutions, trigonometric function solutions, exponential solutions, and rational solutions. The method is straightforward and concise, and its applications are promising.     

Mode i and mixed mode fields with incomplete crack tip plasticity

Plane strain slip line fields, in which plasticity does not fully surround the crack tip have been developed for mode I and mixed mode I\II cracks under contained yielding. Analytical solutions have been assembled using slip line theory for the plastic sectors and semi-infinite wedge solutions for the elastic sectors. These solutions are compared with finite element solutions based on modified boundary layer formulations. The analytical solutions agree well with numerical solutions, and form a family of fields with incomplete plasticity around the crack tip.     

Further improved F-expansion and new exact solutions for nonlinear evolution equations

The improved F-expansion method with a computerized symbolic computation is used to construct the exact traveling wave solutions of four nonlinear evolution equations in physics. As a result, many exact traveling wave solutions are obtained which include new soliton-like solutions, trigonometric function solutions, and rational solutions. The method is straightforward and concise, and it holds promise for many applications.     

非均质流固耦合介质轴对称动力问题时域解

将地基视为流固两相介质并考虑其非均质成层特性，推导了多层地基动力问题时域解．文中首先建立了一组解耦的两相介质动力控制方程；而后利用Ｌａｐｌａｃｅ－Ｈａｎｋｅｌ变换推导了单层地基象空间初参数解答；再利用初参数法及传递矩阵技术导出了任意多层地基瞬态解的一般解析算式．本文获得的解答可方便地退化为现有理想弹性介质的解答     

Planar multiple-contact problems subject to unilateral and bilateral kinetic constraints with static Coulomb friction

This paper is concerned with contact problems. A planar multiple-contact problem subject to unilateral and bilateral kinetic constraints with static friction is studied using the complementarity method. First, this paper discusses the one-to-one correspondence of solutions of the contact problems of concern and of the corresponding complementarity models. An enhanced complementarity model is proposed by adding missed tangential acceleration constraints into previous complementarity models. Solutions of the proposed complementarity model and solutions of the contact problem are proven to exhibit one-to-one correspondence, which may not be guaranteed in the previous complementarity models. Then, this paper applies linear complementarity theory to investigate the properties of the solutions of the proposed complementarity model. For both unilaterally constrained contact problems and bilaterally constrained contact problems, the existence of solutions and boundedness of solutions are proven. Sufficient conditions for the uniqueness of solutions and finiteness of the number of solutions are also provided. Several numerical examples are given to show the non-uniqueness of solutions or the infiniteness of the number of solutions. Such phenomena demonstrate the non-smoothness of the contact problems discussed herein.     

Resonant solutions and breathers to the BKP equation

In this paper, we derive resonant and breather solutions from multi-soliton solutions of the B-type Kadomtsev–Petviashvili (BKP) equation of fourth order via the Hirota bilinear method. We first discuss N-soliton solutions of the BKP equation and use the linear superposition principle to generate N-resonant solutions. Subsequently, we construct complexiton and breather solutions and finally, study the dynamics of some selected solutions with the aid of 3D plots, contour plots and density plots.

     

Large amplitude non-linear oscillations of a general conservative system

This paper presents new, approximate analytical solutions to large-amplitude oscillations of a general, inclusive of odd and non-odd non-linearity, conservative single-degree-of-freedom system. Based on the original general non-linear oscillating system, two new systems with odd non-linearity are to be addressed. Building on the approximate analytical solutions of odd non-linear systems developed by the authors earlier, we construct the new approximate analytical solutions to the original general non-linear system by combinatory piecing of the approximate solutions corresponding to, respectively, the two new systems introduced. These approximate solutions are valid for small as well as large amplitudes of oscillation for which the perturbation method either provides inaccurate solutions or is inapplicable. Two examples with excellent approximate analytical solutions are presented to illustrate the great accuracy and simplicity of the new formulation.     

Classification of Codimension-One Riemann Solutions

We investigate solutions of Riemann problems for systems of two conservation laws in one spatial dimension. Our approach is to organize Riemann solutions into strata of successively higher codimension. The codimension-zero stratum consists of Riemann solutions that are structurally stable: the number and types of waves in a solution are preserved under small perturbations of the flux function and initial data. Codimension-one Riemann solutions, which constitute most of the boundary of the codimension-zero stratum, violate structural stability in a minimal way. At the codimension-one stratum, either the qualitative structure of Riemann solutions changes or solutions fail to be parameterized smoothly by the flux function and the initial data. In this paper, we give an overview of the phenomena associated with codimension-one Riemann solutions. We list the different kinds of codimension-one solutions, and we classify them according to their geometric properties, their roles in solving Riemann problems, and their relationships to wave curves.     

A new method to the (2+1)-dimensional modified KP equation

By means of the auxiliary ordinary differential equation method, we have obtained many solitary wave solutions, periodic wave solutions and variable separation solutions for the (2+1)-dimensional KP equation. Using a mixed method, many exact solutions have been obtained.     

Invariant submodels of the Westervelt model with dissipation

We study three-dimensional Westervelt model of nonlinear hydroacoustics with dissipation. We received all its invariant submodels. With the help of invariant solutions, we explored some wave processes, specifying their physical meaning. The boundary value problems describing these processes are reduced to the nonlinear integro-differential equations. We established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Also we considered the invariant solutions of rank 2 and 3. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters.     

The analyticity of solutions of the Stefan problem

The Stefan problem of a semi-infinite body with arbitrarily prescribed initial and boundary conditions is studied. One of the objectives of the paper is to investigate the analyticity of the solutions. For this purpose, the prescribed initial and boundary conditions are considered to be series of fractional powers of their arguments. It is found that the exact solutions of the problem for various forms of the initial and boundary conditions can be established in series of parabolic cylinder functions and time t. Existence and convergence of the series solutions are studied and proved. The present solutions include the known exact solutions as special cases. On the basis of the present solutions, the question of the analyticity of solutions of the Stefan problem, raised by Rubinstein in his book, can be answered. Conditions for analyticity of the solutions with various initial and boundary conditions are fully discussed.