Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions

A Darboux transformation for the Drinfeld–Sokolov–Satsuma–Hirota system of coupled equations is constructed with the aid of gauge transformations between the Lax pairs. As an application, several types of solutions of the Drinfeld–Sokolov–Satsuma–Hirota system are obtained, including soliton solutions, periodic solutions, rational solutions and others.     

CTE Solvability,Exact Solutions and Nonlocal Symmetries of the Sharma–Tasso–Olver Equation

In this letter,we prove that the STO equation is CTE solvable and obtain the exact solutions of solitons fission and fusion. We also provide the nonlocal symmetries of the STO equation related to CTE. The nonlocal symmetries are localized by prolonging the related enlarged system.     

Symmetries of the Free Schrödinger Equation

An algorithm is proposed for studying the symmetry properties of equations used in theoretical and mathematical physics. The application of this algorithm to the free Schrödinger equation permits one to establish that, in addition to the known Galilei symmetry, the free Schrödinger equation possesses also relativistic symmetry in some generalized sense. This property of the free Schrödinger equation provides an extension of the equation into the relativistic domain of the free particle motion under study.     

Rational Solutions of the Master Symmetries¶of the KdV Equation

In this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for KdV. The result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master-symmetry KdV hierarchy are bispectral potentials for the Schr?dinger operator. By bispectral potentials we mean that the corresponding Schr?dinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable. This complements certain results of Airault–McKean–Moser [4], Duistermaat–Grünbaum [10], and Magri–Zubelli [40]. As a consequence of bispectrality, the rational solutions of the master symmetries turn out to be solutions of a (generalized) string equation. Received: 28 January 1999 / Accepted: 22 October 1999     

Inverse Scattering Transform of the Coupled Sasa–Satsuma Equation by Riemann–Hilbert Approach

The inverse scattering transform of a coupled Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Firstly, the spectral analysis is performed for the coupled Sasa–Satsuma equation, from which a Riemann–Hilbert problem is formulated. Then the Riemann–Hilbert problem corresponding to the reflection-less case is solved.As applications, multi-soliton solutions are obtained for the coupled Sasa–Satsuma equation. Moreover, some figures are given to describe the soliton behaviors, including breather types, single-hump solitons, double-hump solitons, and two-bell solitons.     

Infinitely Many Symmetries of Konopelchenko-Dubrovsky Equation

A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky （KD） equation in 2＋1 space dimensions is given by using a direct method called formal function series method presented by Lou. These symmetries constitute an infinite-dimensional generalized w∞ algebra.     

A New Solution to the Hirota–Satsuma Coupled KdV Equations by the Dressing Method

The Hirota–Satsuma coupled KdV equations associated 2×2 matrix spectral problem is discussed by the dressing method,which is based on the factorization of integral operator on a line into a product of two Volterra integral operators.A new solution is obtained by choosing special kernel of integral operator.     

A Note on Symmetries and Generalized W∞ Algebra of the Modified KP Equation

Some remarks on the paper Symmetries and generalized W algebra of the modified KP equation (S. Y. Lou and G. J. Ni, Lett. Math. Phys. 34 (1995), 327–331) are given. It is pointed out that if we consider the inverse operator of a differential operator to be a linear operator, the vector fields nv(f) defined in the above Letter are not certain to be symmetries of the modified KP equation under consideration.     

Trajectory equation of a lump before and after collision with other waves for generalized Hirota–Satsuma–Ito equation

Based on the Hirota bilinear and long wave limit methods, the hybrid solutions of m-lump with n-soliton and nbreather wave for generalized Hirota–Satsuma–Ito(GHSI) equation are constructed. Then, by approximating solutions of the GHSI equation along some parallel orbits at infinity, the trajectory equation of a lump wave before and after collisions with n-soliton and n-breather wave are studied, and the expressions of phase shift for lump wave before and after collisions are given. Furthermore, ...     

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

In this paper, two types of fractional nonlinear equations in Caputo sense, time-fractional Newell–Whitehead equation(FNWE) and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS), are investigated by means of the q-homotopy analysis method(q-HAM). The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions. Due to the presence of the auxiliary parameter h in this method, just a few terms of the series solution are required in order to obtain better approximation. For the sake of visualization, the numerical results obtained in this paper are graphically displayed with the help of Maple.     

Optical solitons and complexitons of the Schrödinger–Hirota equation

The Schrödinger–Hirota equation governs the propagation of optical solitons in a dispersive optical fiber. In this paper, this equation will be solved by the ansatz method for bright and dark 1-soliton solution. The power law nonlinearity will be assumed. By using the tanh method, some additional solutions will be derived. Finally, the numerical simulations will be given.     

Reducible Connections and Non-local Symmetries of the Self-dual Yang–Mills Equations

We construct the most general reducible connection that satisfies the self-dual Yang–Mills equations on a simply-connected, open subset of flat ${\mathbb{R}^4}$ . We show how all such connections lie in the orbit of the flat connection on ${\mathbb{R}^4}$ under the action of non-local symmetries of the self-dual Yang–Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang–Mills equations that are analogous to harmonic maps of finite type.     

Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation

We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I Equation obtained by the inverse spectral method belongs to the Sobolev space H^k(R²) for k 0, under the assumption that the initial datum is a small Schwartz function. This solution is shown to be the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation.     

Combinatorics and Formal Geometry of the Maurer–Cartan Equation

We give a general treatment of the Maurer–Cartan equation in homotopy algebras and describe the operads and formal differential geometric objects governing the corresponding algebraic structures. We show that the notion of Maurer–Cartan twisting is encoded in certain automorphisms of these universal objects.     

Coupled Nonlinear Schrodinger Equation： Symmetries and Exact Solutions

The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger （CNLS） equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.