Algebro-Geometric Solutions of the TD Hierarchy

Resorting to the Lenard recursion scheme, we derive the TD hierarchy associated with a 2?×?2 matrix spectral problem and establish Dubrovin-type equation in terms of the introduced elliptic variables. Based on the theory of algebraic curve, all the flows associated with the TD hierarchy are straightened under the Abel-Jacobi coordinates. An algebraic function ?, also called the meromorphic function, carrying the data of the divisor is introduced on the underlying hyperelliptic curve $\mathcal {K}_{n}$ . The known zeros and poles of ? allow to find theta function representations for ? by referring to Riemann’s vanishing theorem, from which we obtain algebro-geometric solutions for the entire TD hierarchy with the help of asymptotic expansion of ? and its theta function representation.     

Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS2 hierarchy.     

Algebro-Geometric Solutions with Characteristics of a Nonlinear Partial Differential Equation with Three-Potential Functions

With the help of a simple Lie algebra, an isospectral Lax pair, whose feature presents decomposition of element (1, 2) into a linear combination in the temporal Lax matrix, is introduced for which a new integrable hierarchy of evolution equations is obtained, whose Hamiltonian structure is also derived from the trace identity in which contains a constant γ to be determined. In the paper, we obtain a general formula for computing the constant γ. The reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions, the mKdV equation and a generalized linear KdV equation. The algebro-geometric solutions (also called finite band solutions) of the generalized nonlinear heat equation are obtained by the use of theory on algebraic curves. Finally, two kinds of gauge transformations of the spatial isospectral problem are produced.     

New Optical Soliton Solutions of the Perturbed Fokas-Lenells Equation

In this article, we employ the perturbed Fokas-Lenells equation(FLE), which represents recent electronic communications. The Riccati-Bernoulli Sub-ODE method which does not depend on the balance rule is used for thefirst time to obtain the new exact and solitary wave solutions of this equation. This technique is direct, effective and reduces the large volume of calculations.     

Quasiperiodic Solutions of the Heisenberg Ferromagnet Hierarchy

We present quasi-periodic solutions in terms of Riemann theta functions of the Heisenberg ferromagnet hierarchy by using algebro-geometric method. Our main tools include algebraic curve and Riemann surface, polynomial recursive formulation and a special meromorphic function.     

Nonequivalent Similarity Reductions and Exact Solutions for Coupled Burgers-Type Equations

Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.     

Stability Analysis of Solitary Wave Solutions for Coupled and(2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications

The searching exact solutions in the solitary wave form of non-linear partial differential equations(PDEs play a significant role to understand the internal mechanism of complex physical phenomena. In this paper, we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the(2+1)-dimensional cubic Klein-Gordon(K-G) equation. The Klein-Gordon equation are relativistic version of Schr¨odinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which severa solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions o PDEs arise in mathematical physics.     

A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.     

Exact Solutions of Ablowitz-Ladik Hierarchy with Self-Consistent Sources Revisited and Reductions

In the paper, Ablowitz-Ladik hierarchy with new self-consistent sources is investigated. First the source in the hierarchy is described as φ_nφ_n+1, where φ_n is related to the Ablowitz-Ladik spectral problem, instead of the corresponding adjoint spectral problem. Then by means of the inverse scattering transform, the multi-soliton solutions for the hierarchy are obtained. Two reductions to the discrete mKdV and nonlinear Schrödinger hierarchies with self-consistent sources are considered by using the uniqueness of the Jost functions, as well as their N-soliton solutions.     

Multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas-Lenells equation

In this letter, we investigate multisoliton solutions with even numbers and its generated solutions for nonlocal Fokas-Lenells equation over a nonzero background. First, we obtain 2n-soliton solutions with a nonzero background via n-fold Darboux transformation, and find that these soliton solutions will appear in pairs. Particularly, 2n-soliton solutions consist of n ‘bright' solitons and n ‘dark' solitons. This phenomenon implies a new form of integrability: even integrability. Then interactions between solitons with even numbers and breathers are studied in detail. To our best knowledge, a novel nonlinear superposition between a kink and 2n-soliton is also generated for the first time. Finally, interactions between some different smooth positons with a nonzero background are derived.     

Self-Similar Solutions of the Boltzmann Equation and Their Applications

We consider a class of solutions of the Boltzmann equation with infinite energy. Using the Fourier-transformed Boltzmann equation, we prove the existence of a wide class of solutions of this kind. They fall into subclasses, labelled by a parameter a, and are shown to be asymptotic (in a very precise sense) to the self-similar one with the same value of a (and the same mass). Specializing to the case of a Maxwell-isotropic cross section, we give evidence to the effect that the only self-similar closed form solutions are the BKW mode and the two solutions recently found by the authors. All the self-similar solutions discussed in this paper are eternal, i.e., they exist for –<t<, which shows that a recent conjecture cannot be extended to solutions with infinite energy. Eternal solutions with finite moments of all orders, and different from a Maxwellian, are also studied. It is shown that these solutions cannot be positive. Moreover all such solutions (partly negative) must be asymptotically (for large negative times) close to the exact eternal solution of BKW type.     

Exact Solutions of a Coupled Burgers System

The exact solutions of a new coupled Burgers system are studied in three different ways. The first type of solutions are found thanks to the coupled Burgers system possessing a simple single Burgers reduction. The second type of multiple soliton solutions are revealed via the decouple procedure. The third type of exact solutions are found by means of a prior ansatz and solutions of the heat conduction equation. Two different kinds of soliton fission phenomena of the model are discovered and a special type of completely elastic soliton collision without phase shift of the model is also displayed.     

Dynamics of Analytical Matter-Wave Solutions in Three-Dimensional Bose–Einstein Condensates with Two- and Three-Body Interactions

Using the F-expansion method we present analytical matter-wave solutions to Bose-Einstein condensates with two- and three-body interactions through the generalized three-dimensional Gross-Pitaevskii equation with time- dependent coefficients, for the periodically time-varying interactions and quadratic potential strength. Such solutions exist under certain conditions, and impose constraints on the functions describing potential strength, nonlinearities, and gain （loss）. Various shapes of analytical matter-wave solutions which have important applications of physical interest are s~udied in details.     

Huygens' Principle,Dirac Operators,and Rational Solutions of the AKNS Hierarchy

We prove that rational solutions of the AKNS hierarchy of the form q=σ/τ and r=ρ/τ, where (σ,τ,ρ) are certain Schur functions, naturally yield Dirac operators of strict Huygens' type, i.e., the support of their fundamental solutions is the surface of the light-cone. This strengthens the connection between the theory of completely integrable systems and Huygens' principle by extending to the Dirac operators and the rational solutions of the AKNS hierarchy a classical result of Lagnese and Stellmacher concerning perturbations of wave operators. Mathematics Subject Classifications (2000) 37K10, 35Qxx, 35B40.     

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.     

Jacobian Elliptic Function Method and Solitary Wave Solutions for Hybrid Lattice Equation

In this paper, we have successfully extended the Jacobian elliptic function expansion approach to nonlinear differential-difference equations. The Hybrid lattice equation is chosen to illustrate this approach. As a consequence, twelve families of Jacobian elliptic function solutions with different parameters of the Hybrid lattice equation are obtained. When the modulus m→1 or 0, doubly-periodic solutions degenerate to solitonic solutions and trigonometric function solutions, respectively.     

New Periodic Wave Solutions, Localized Excitations and Their Interaction Properties for （3＋1）-Dimensional Jimbo-Miwa Equation

Based on the multi-linear variable separation approach, a class of exact, doubly periodic wave solutions for the （3＋1）-dimensional Jimbo-Miwa equation is analytically obtained by choosing the Jacobi elliptic functions and their combinations. Limit cases are considered and some new solitary structures （new dromions） are derived. The interaction properties of periodic waves are numerically studied and found to be inelastic. Under long wave limit, two sets of new solution structures （dromions） are given. The interaction properties of these solutions reveal that some of them are completely elastic and some are inelastic.     

Symmetry, Reductions and New Exact Solutions of ANNV Equation Through Lax Pair

In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.     

Darboux Coverings and Rational Reductions of the KP Hierarchy

We use the method of Darboux coverings to discuss the invariant submanifolds of the KP equations presented as conservation laws in the space of monic Laurent series in the spectral parameter (the space of the Hamiltonian densities). We identify a special class of these submanifolds with the rational invariant submanifolds entering matrix models of two-dimensional gravity recently characterized by Dickey and Krichever. Four examples of the general procedure are provided.