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In this study, we focus on investigating a complex modified short pulse (CSP) equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Riemann-Hilbert problem for the initial-value problem of the CSP equation is established. The one-soliton solutions and breather-type solutions of the CSP equation are further given by solving the Riemann-Hilbert problem. [ABSTRACT FROM AUTHOR]

Recently, Fan, Gao and Liu proposed a kind of rotation-two-component Camassa–Holm system. In this paper, we investigate whether the rotation-two-component Camassa–Holm system admits peakon-delta weak solutions in distribution sense. As special reductions, all peakon solutions for generalized Dullin–Gottwald–Holm system, two-component Camassa–Holm system, Dullin–Gottwald–Holm equation and Camassa–Holm equation are recovered from the corresponding results of rotation-two-component Camassa–Holm system. [ABSTRACT FROM AUTHOR]

In this paper, we present the construction of quasiperiodic solutions of the p th equation in the modified Kadomtsev–Petviashvili hierarchy by seeking compatible solutions of the 1th and the p th equations in the derivative nonlinear Schrödinger hierarchy. [ABSTRACT FROM AUTHOR]

Without solving the equations, we show that soliton solutions of the relativistic Lotka–Volterra hierarchy can be derived from those of the Lotka–Volterra hierarchy. The proof is based on the reductions of a discrete spectral problem. [ABSTRACT FROM AUTHOR]

This paper aims at providing theta function representation of all algebro-geometric solutions and related quantities for the Fokas–Olver–Rosenau–Qiao (FORQ, sometimes also called the modified Camassa–Holm (MCH) in the literature) hierarchy and studying their algebro-geometric initial value problem. Our main tools include the polynomial recursion formalism to derive the FORQ hierarchy, the hyperelliptic curve K n of arithmetic genus n , the Baker–Akhiezer functions, the meromorphic function ϕ , the Dubrovin-type equations for auxiliary divisors and the associated trace formulas. With the help of these tools, the explicit theta function representations of the Baker–Akhiezer functions, the meromorphic function and the algebro-geometric solutions are obtained for the entire FORQ hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we present a unified formulae for explicit self-similar solutions of the Euler/Navier–Stokes–Korteweg equations arising in the modeling of capillary fluids. The technique used here is to reduce Euler/Navier–Stokes–Korteweg equations into a series of solvable ordinary differential equations by making use of multi-dimensional self-similar ansatz and variable separation method. [ABSTRACT FROM AUTHOR]

We show that, under an irrotational condition, there exists an n -dimensional Hopf–Cole transformation between the n -dimensional Burgers system and an n -dimensional heat equation. Further, as application of the Hopf–Cole transformation, two kinds of physically interesting exact solutions for the n -dimensional Burgers equations are found. In the first kind of solutions, the velocity fields are topological solitons. In the second kind of solutions, velocity fields are all multiple fusion soliton solutions. [ABSTRACT FROM AUTHOR]

We apply the ∂ ¯ -dressing method to study nonlocal modified Korteweg-de Vries (nonlocal mKdV) equation. A spatial and a time singular spectral problems associated with nonlocal mKdV equation are derived from a local 2 × 2 matrix ∂ ¯ -equation via two linear constraint equations. A nonlocal mKdV hierarchy is proposed by using recursive operator. And the conservation laws of the nonlocal mKdV are derived by the temporal linear spectral problem. The N -solitions of the nonlocal mKdV equation are constructed still based the ∂ ¯ -equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained. The results on the mKdV equation can be recovered from the conclusion given above as special reductions. [ABSTRACT FROM AUTHOR]

Abstract: For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found. [Copyright &y& Elsevier]

Highlights: [•] We present a Riemann–Hilbert problem formalism for the initial–boundary value problem of the three-wave equation. [•] The residue conditions of matrix function M in the Riemann–Hilbert problem is obtained. [•] The explicit jump matrix J in the Riemann–Hilbert problem is constructed. [Copyright &y& Elsevier]

Abstract: A linear superposition principle of exponential traveling waves is analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of -soliton solutions formed by linear combinations of exponential traveling waves. Applications are made for the 3+1 dimensional KP, Jimbo–Miwa and BKP equations, thereby presenting their particular -wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated -wave solutions, and a few illustrative examples are presented, together with an algorithm using weights. [Copyright &y& Elsevier]

Abstract: Binary Bell polynomials are extended to systematically construct bilinear formalism, bilinear Bäcklund transformations, Lax pairs and infinite conservation laws of the nonisospectral and variable-coefficient KdV equation in a quick and natural way. Moreover, the infinite conservation laws are local and obtained through directly decoupling binary Bell polynomials. [ABSTRACT FROM AUTHOR]

Abstract: Based on the Hirota bilinear method and the Riemann theta function, a straightforward way is shown to construct quasi-periodic wave solutions of supersymmetric equations. The resulting theory is applied to the supersymmetric modified Korteweg–de Vries equation. Further, we analyze the asymptotic properties of the solutions and give their asymptotic relations between the quasi-periodic wave solutions and the soliton solutions. [Copyright &y& Elsevier]

Abstract: A direct and unifying scheme for disclosure of periodic wave solutions of both nonlinear differential and difference equations is presented. The scheme is based on Hirota''s bilinear form and certain Riemann theta function formulae. The relations between the periodic wave solutions and soliton solutions are rigorously established. [Copyright &y& Elsevier]

Abstract: In the present Letter, by using Hirota bilinear method and Riemann theta function, the quasi-periodic wave solutions are constructed for the -dimensional Modified Nizhnik–Novikov–Vesselov equation, and we show asymptotic properties of the quasi-periodic wave solutions. [Copyright &y& Elsevier]

Abstract: In this Letter, we propose a supersymmetric KdV–Sawada–Kotera–Ramani equation. Based on a super-Riemann theta function, we devise a lucid and straightforward way for explicitly constructing a quasi-periodic wave solution of the supersymmetric KdV–Sawada–Kotera-Ramani equation. In addition, a one-soliton solution is obtained as a limiting case of the periodic wave solution under small amplitude. Indeed different from the purely bosonic case, the quasi-periodic wave observed shows that there is an “influencing band” among the waves under the presence of the Grassmann variable. The waves are symmetric about the band but collapse along with the band. Furthermore, the amplitudes of the waves increase as the waves move away from the band. [Copyright &y& Elsevier]

Abstract: Staring from a new spectral problem, a hierarchy of the soliton equations is derived. It is shown that the associated hierarchies are infinite-dimensional integrable Hamiltonian systems. By the procedure of nonlinearization of the Lax pairs, the integrable decomposition of the whole soliton hierarchy is given. Further, we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of Lie algebras. [Copyright &y& Elsevier]

Abstract: By introducing a discrete spectral problem, we derive a lattice hierarchy which is integrable in Liouville''s sense and possesses a multi-Hamiltonian structure. It is show that the discrete spectral problem converges to the well-known AKNS spectral problem under a certain continuous limit. In particular, we construct a sequence of equations in the lattice hierarchy which approximates the AKNS hierarchy as a continuous limit. [Copyright &y& Elsevier]

Abstract: By embedding a free function into a compatible zero curvature equation, we enlarge the original differential-difference hierarchy into a new hierarchy with the free function which still admits zero curvature representation. The new hierarchy not only includes the original hierarchy, but also the well-known relativistic Toda hierarchy and the Volterra hierarchy as special reductions by properly choosing the free function. Infinitely many conservation laws and Darboux transformation for a representative differential-difference system are constructed based on its Lax representation. The exact solutions follow by applying the Darboux transformation. [Copyright &y& Elsevier]

Abstract: Starting from a new discrete spectral problem, the corresponding positive and negative hierarchies of nonlinear lattice equations are proposed. It is shown that the both lattice soliton hierarchies possess the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, we construct two integrable coupling systems for the positive hierarchy through enlarging Lax pair method. [Copyright &y& Elsevier]

Abstract: By constructing a type of antisymmetric matrix Lie algebra, a way to generate enlarging integrable systems is presented. The such integrable hierarchies are not integrable couplings, specially, their corresponding Hamiltonian structure can be obtained by the trace identity. As an application example, the enlarging integrable system and the Hamiltonian structure of the AKNS hierarchy are given. Finally, the explicit relations expressed by the formulae among the solutions of the stationary zero curvature equations are obtained. [Copyright &y& Elsevier]

Abstract: We present a new basis of the loop algebra which is devote to setting up a new isospectral problem. By using Tu scheme, a new Liouville integrable hierarchy of soliton equations with two arbitrary parameters, which possesses the bi-Hamiltonian structure, is worked out. As reduction cases, it is shown that the hierarchy can be decomposed into the well-known AKNS hierarchy and the BPT hierarchy by taking different values of the parameters respectively. We call the hierarchy as an unified expression of the AKNS and the BPT hierarchies. [Copyright &y& Elsevier]

Abstract: Based on two foundational subalgebras of the Lie algebra , a few expanding higher-dimensional Lie algebras are presented to generate four integrable couplings of a soliton equation hierarchy. The Hamiltonian structure for one of them is obtained by using the quadratic-form identity. The classification of the Lie algebras is also given. Moreover, a decomposition of one higher-dimensional Lie algebra is demonstrated to produce a q-integrable expanding hierarchy for generalized q-form KdV hierarchy. [Copyright &y& Elsevier]

Abstract: With the help of two semi-direct sum Lie algebras, an efficient way to construct discrete integrable couplings is proposed. As its applications, the discrete integrable couplings of the Toda-type lattice equations are obtained. The approach can be devoted to establishing other discrete integrable couplings of the discrete lattice integrable hierarchies of evolution equations. [Copyright &y& Elsevier]

Abstract: A scheme for deducing Hamiltonian structures of the higher-dimensional hierarchies of evolution equations is presented which is devoting to obtaining the Hamiltonian structures of integrable coupling of the Jaulent–Miodek hierarchy. [Copyright &y& Elsevier]

In this paper, we explore more applications of extended tanh method to some special nonlinear equations. Such equations are not of differential polynomial form so that cannot be directly dealt with by tanh method or extended tanh method. As examples, we apply a recently proposed extended tanh method to build some new explicit solutions for (2+1)-dimensional sine-Gordon equation, Dodd–Bullough–Mikhailov equation and coupled Schro¨dinger-KdV equation. [Copyright &y& Elsevier]

The Jacobi elliptic function method with symbolic computation is extended to special-type nonlinear equations for constructing their doubly periodic wave solutions. Such equations cannot be directly dealt with by the method and require some kinds of pre-possessing techniques. It is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions. The different Jacobi function expansions may lead to new Jacobi doubly periodic wave solutions, triangular periodic solutions and soliton solutions. In addition, as an illustrative sample, the properties for the Jacobi doubly periodic wave solutions of the coupled Schro¨dinger–KdV equation are shown with some figures. [Copyright &y& Elsevier]

We present a simple and effective transformation that decompose the solution of a coupled system into solving a set of algebraic equations and a first order ordinary differential equation of a certain unknown function. This method not only gives us a clear relation between unknown functions for a nonlinear coupled system, but also easily provides us with a series of new and more general travelling wave solutions in terms of special functions such as hyperbolic, rational, triangular, Weierstrass and Jacobi elliptic double periodic functions. [Copyright &y& Elsevier]

Based on the idea of the homogeneous balance (HB) method, we study the Ba¨cklund transformation and similarity reductions of general variable coefficient KdV equation. It is found that the corresponding results are coincide with those by Weiss–Tabor–Carnevale (WTC) method and Clarkson–Kruskal (CK) method, respectively. The close connections among the HB, WTC and CK methods are exposured in theory. In the meantime, a Lax pair, a symmetry, two conservation laws and a analytic solution for the general variable coefficient KdV equation are given. [Copyright &y& Elsevier]

We apply the Dbar-dressing method to study a Gerdjikov–Ivanov (GI) equation with nonzero boundary at infinity. A spatial and a time spectral problem associated with GI equation are derived with a asymptotic expansion method. The N-soliton solutions of the GI equation are constructed based the Dbar-equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained. • The ∂ ̄ -dressing method to study the GI equation with nonzero boundary. • A spatial and a time spectral problem associated with GI equation are derived. • N-solitons of the GI equation with nonzero boundary are constructed. [ABSTRACT FROM AUTHOR]

Starting from a local 2 × 2 matrix ∂ ¯ -equation with non-normalization boundary conditions, the spatial and time spectral problems associated with the Kundu-Eckhaus (KE) equation are derived via two linear constraint equations. The gauge equivalence among the KE equation, NLS equation and Heisenberg chain equation are given. A KE hierarchy with source is proposed by using recursive operator. The N -solitions of the KE equation are constructed based the ∂ ¯ -equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained. [ABSTRACT FROM AUTHOR]

We apply the ∂ ̄ -dressing method to study a coupled Gerdjikov–Ivanov (GI) equation. Compared with results on the classical coupled GI equation, more general spatial and a time singular spectral problems associated with GI equation are derived from a local 2 × 2 matrix ∂ ̄ -equation via two linear constraint equations. A more general GI hierarchy with source is proposed by using recursive operator. The N -solitons of the coupled GI equation are constructed still based the ∂ ̄ -equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained. The results on the GI equation can be recovered from the conclusion given above as special reductions. [ABSTRACT FROM AUTHOR]

We make use of an extended tanh method with symbolic computation to find the soliton solutions, triangular periodic solutions and rational solutions of a new coupled MKdV system in an unified way. The key idea of this method is to take full advantage of a Riccati equation involving a parameter and use its solutions to replace the tanh-function in the tanh method. It is quite interesting that the sign of the parameter can be used to exactly classify the amount and pattern of such travelling wave solutions. [Copyright &y& Elsevier]

We make use of Jacobi elliptic functions to construct new double periodic wave solutions for two generalized Hirota–Satsuma coupled KdV systems. It is shown that these solutions exactly degenerate to the well-known soliton solutions at a limit condition. [Copyright &y& Elsevier]