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.     

Evolution Property of Multisoliton Excitations for a Higher-Dimensional Coupled Burgers System

By means of the standard truncated Painlevé expansion and a special Bäcklund transformation, the higher-dimensional coupled Burgers system (HDCB) is reduced to a linear equation, and an exact multisoliton excitation is derived. The evolution properties of the multisoliton excitation are investigated and some novel features or interesting behaviors are revealed. The results show that after interactions for dromion-dromion, solitoff-solitoff, and solitoff-dromion, they are combined with some new types of localized structures, which are similar to classic particles with completely nonelastic behaviors.     

New Rational Form Solutions to Coupled Nonlinear Wave Equations

The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.     

New Rational Form Solutions to Coupled Nonlinear Wave Equations

The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.     

Complexiton Solutions of a Special Coupled mKdV System

For a special coupled mKdV system, which can be derived from a two-layer fluid model, Hirota＇s bilinear direct method is used to construct and yield the complexiton solutions. The detailed physical properties of complexitons are filrther illustrated graphically.     

Solitary Wave Solutions for the Coupled Ito System and a Generalized Hirota-Satsuma Coupled KdV System

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Rational Solutions for the Fokas System

Fokas system is the simplest(2+1)-dimensional extension of the nonlinear Schrodinger equation(Eq.(2),Inverse Problems 10(1994) L19-L22).By using the bilinear transformation method,general rational solutions for the Fokas system are given explicitly in terms of two order-N determinants T_n(n = 0,1) whose elements m_(i,j)~(n)(n = 0,1;1≤i,j≤N)are involved with order-n_i and order-n_j derivatives.When N = 1,three kinds of rational solution,i.e.,fundamental lump and fundamental rogue wave(RW) with n_1 = 1,and higher-order rational solution with n_1 2,are illustrated by explicit formulas from T_n(n = 0,1) and pictures.The fundamental RW is a line RW possessing a line profile on(x,y)-plane,which arises from a constant background with at t 0 and then disappears into the constant background gradually at t 0.The fundamental lump is a traveling wave,which can preserve its profile during the propagation on(x,y)-plane.When N ≥2 and n_1 =n_2=...=n_n = 1,several specific multi-rational solutions are given graphically.     

Rational Solutions for the Fokas System

Fokas system is the simplest (2+1)-dimensional extension of the nonlinear Schrödinger equation (Eq. (2), Inverse Problems 10 (1994) L19-L22). By using the bilinear transformation method, general rational solutions for the Fokas system are given explicitly in terms of two order-N determinants τn (n = 0, 1) whose elements m_i,j⁽ⁿ⁾ (n = 0, 1; 1 ≤ i, j ≤ N) are involved with order-n_i and order-n_j derivatives. When N = 1, three kinds of rational solution, i.e., fundamental lump and fundamental rogue wave (RW) with n₁ = 1, and higher-order rational solution with n₁ ≥ 2, are illustrated by explicit formulas from τn (n = 0, 1) and pictures. The fundamental RW is a line RW possessing a line profile on (x, y)-plane, which arises from a constant background with at t << 0 and then disappears into the constant background gradually at t >> 0. The fundamental lump is a traveling wave, which can preserve its profile during the propagation on (x, y)-plane. When N ≥ 2 and n₁ = n₂ = ··· = n_N = 1, several specific multi-rational solutions are given graphically.     

Integrable Couplings of the Coupled Burgers Hierarchy

In this letter, a new loop algebra G is constructed, from which a new isospectral problem is established. It follows that integrable couplings of the well-known coupled Burgers hierarchy are obtained.     

Integrable Couplings of the Coupled Burgers Hierarchy

In this letter, a new loop algebra G is constructed, from which a new isospectral problem is established. It follows that integrable couplings of the well-known coupled Burgers hierarchy are obtained.     

Markovian Solutions of Inviscid Burgers Equation

For solutions of (inviscid, forceless, one dimensional) Burgers equation with random initial condition, it is heuristically shown that a stationary Feller–Markov property (with respect to the space variable) at some time is conserved at later times, and an evolution equation is derived for the infinitesimal generator. Previously known explicit solutions such as Frachebourg–Martin's (white noise initial velocity) and Carraro–Duchon's Lévy process intrinsic-statistical solutions (including Brownian initial velocity) are recovered as special cases.     

Bell Polynomial Approach and N-Soliton Solutions for a Coupled KdV-mKdV System

In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.     

Bell Polynomial Approach and N-Soliton Solutions for a Coupled KdV-mKdV System

In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.     

Similarity Reductions and Exact Solutions of a Coupled Korteweg de Vries System

For a special coupled Korteweg de Vries （KdV） system, its similarity solutions and reduction equations are obtained by the Clarkson and Kruskal＇s direct method. In addition, its new explicit soliton solutions and traveling wave solutions are found by the deformation and mapping method.     

A Study of Brane Solutions in D-Dimensional Coupled Gravity System

We use only the equation.of motion for an interacting system of gravity, dilaton and antisymmetric tensor to study the soliton solutions by making use of a Poincaré-invariant ansatz. We show that the system of equations is completely integrable and the solution is unique with appropriate boundary conditions. Some new classes of solutions are also given explicitly.     

New Rational Solutions for Relativistic Discrete Toda Lattice System

In the present work, we examine the soliton excitations in the relativistic Toda lattice model using the rotational expansion method, where the coupling between the lattice sites is varied. For specific choices of the coupling strength we proceed to analyze the nonlinear wave excitations arising in the model which are found to be dark, singular and periodic solitary wave profiles. These solitary wave profiles are admitted to show possible modulation in its amplitude.     

New Rational Solutions for Relativistic Discrete Toda Lattice System

In the present work, we examine the soliton excitations in the relativistic Toda lattice model using the rotational expansion method, where the coupling between the lattice sites is varied. For specific choices of the coupling strength we proceed to analyze the nonlinear wave excitations arising in the model which are found to be dark, singular and periodic solitary wave profiles. These solitary wave profiles are admitted to show possible modulation in its amplitude.     

New Exact Solutions and Localized Structures for （3＋1）-Dimensional Burgers System

With an extended mapping approach and a linear variable separation method, new families of variable separation solutions （including solitary wave solutions, periodic wave solutions, and rational function solutions） with arbitrary functions for （3＋1）-dimensionai Burgers system is derived. Based on the derived excitations, we obtain some novel localized coherent structures and study the interactions between solitons.     

The Neumann Type Systems and Algebro-Geometric Solutions of a System of Coupled Integrable Equations

A system of (1+1)-dimensional coupled integrable equations is decomposed into a pair of new Neumann type systems that separate the spatial and temporal variables for this system over a symplectic submanifold. Then, the Neumann type flows associated with the coupled integrable equations are integrated on the complex tour of a Riemann surface. Finally, the algebro-geometric solutions expressed by Riemann theta functions of the system of coupled integrable equations are obtained by means of the Jacobi inversion.