Complete integrability of coupled KdV and NLS equations

Painlevé analysis is performed for the coupled system of nonlinear partial differential equations consisting of the KdV equation and NLS equation initially studied by Nishikawa. Various possibilities for the constants occurring in the system are explored, paying attention to the integrability property. This equation occurring in the field of plasma physics satisfies all the requirements of Painlevé analysis and can be ascertained to be completely integrable, though no Lax pair is known for it.     

New exact solutions to a system of coupled KdV equations

Using an improved homogeneous balance method, several kinds of exact solutions which include Wang's results are obtained for a system of coupled KdV equations.     

Integrability of coupled KdV equations

The integrability of coupled KdV equations is examined. The simplified form of Hirota’s bilinear method is used to achieve this goal. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation. The resonance phenomenon of each model will be examined.     

The Hamilton Cartan formalism for rth-order Lagrangians and the integrability of the KdV and modified KdV equations

The Hamilton Cartan formalism for rth order Lagrangians is presented in a form suited to dealing with higher-order conserved currents. Noether's Theorem and its converse are stated and Poisson brackets are defined for conserved charges. An isomorphism between the Lie algebra of conserved currents and a Lie algebra of infinitesimal symmetries of the Cartan form is established. This isomorphism, together with the commutativity of the Bäcklund transformations for the KdV and modified KdV equations, allows a simple geometric proof that the infinite collections of conserved charges for these equations are in involution with respect to the Poisson bracket determined by their Lagrangians. Thus, the formal complete integrability of these equations appears as a consequence of the properties of their Bäcklund transformations.It is noted that the Hamilton Cartan formalism determines a symplectic structure on the space of functionals determined by conserved charges and that, in the case of the KdV equation, the structure is the same as that given by Miura et al. [5].     

Exact solutions to a coupled modified KdV equations with non-uniformity terms

The bilinear form of a coupled modified KdV equations with non-uniformity terms is given and a few soliton solutions are obtained. Furthermore, the multisoliton of the coupled system is expressed by Pfaffian.     

Variable separation solutions and interacting waves of a coupled system of the modified KdV and potential BLMP equations

The multilinear variable separation approach (MLVSA) is applied to a coupled modified Korteweg–de Vries and potential Boiti–Leon–Manna–Pempinelli equations, as a result, the potential fields _uy$u_y$ and _vy$v_y$ are exactly the universal quantity applicable to all multilinear variable separable systems. The generalized MLVSA is also applied, and it is found that _uy$u_y$ (_vy$v_y$) is rightly the subtraction (addition) of two universal quantities with different parameters. Then interactions between periodic waves are discussed, for instance, the elastic interaction between two semi-periodic waves and non-elastic interaction between two periodic instantons. An attractive phenomenon is observed that a dromion moves along a semi-periodic wave.     

Applications of F-expansion method to the coupled KdV system

An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by Jacobi elliptic functions for the coupled KdV equations are derived. In the limit cases, the solitary wave solutions and the other type of travelling wave solutions for the system are also obtained.     

Exact solutions for KdV system equations hierarchy

Exact solutions for KdV system equations hierarchy are obtained by using the inverse scattering transform. Exact solutions of isospectral KdV hierarchy, nonisospectral KdV hierarchies and τ$\tau$-equations related to the KdV spectral problem are obtained by reduction. The interaction of two solitons is investigated.     

New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.     

Nonsingular positon and complexiton solutions for the coupled KdV system

Taking the coupled KdV system as a simple example, analytical and nonsingular complexiton solutions are firstly discovered in this Letter for integrable systems. Additionally, the analytical and nonsingular positon–negaton interaction solutions are also firstly found for S-integrable model. The new analytical positon, negaton and complexiton solutions of the coupled KdV system are given out both analytically and graphically by means of the iterative Darboux transformations.     

Lie symmetry analysis and reduction of a new integrable coupled KdV system

In this paper, Lie symmetry is investigated for a new integrable coupled Korteweg--de Vries (KdV) equation system. Using some symmetry subalgebra of the equation system, we obtain five types of the significant similarity reductions. Abundant solutions of the coupled KdV equation system, such as the solitary wave solution, exponential solution, rational solution and polynomial solution, etc. are obtained from the reduced equations. Especially, one type of group-invariant solution of reduced equations can be acquired by means of the Painlev\'e I transcendent function.     

Nonlocal KdV equations

Writing the Hirota-Satsuma (HS) system of equations in a symmetrical form we find its local and new nonlocal reductions. It turns out that all reductions of the HS system are Korteweg-de Vries (KdV), complex KdV, and new nonlocal KdV equations. We obtain one-soliton solutions of these KdV equations by using the method of Hirota bilinearization.     

On integrability of the Szekeres system. I

The Szekeres system is a four-dimensional system of ?rst-order ordinary differential equations with nonlinear but polynomial (quadratic) right-hand side. It can be derived as a special case of the Einstein equations, related to inhomogeneous and nonsymmetrical evolving spacetime. The paper shows how to solve it and ?nd its three global independent ?rst integrals via Darboux polynomials and Jacobi’s last multiplier method. Thus the Szekeres system is completely integrable. Its two-dimensional subsystem is also investigated: we present its solutions explicitly and discuss its behaviour at in?nity.     

Fluctuating solitons of the KdV equations

We introduce a new AKNS three-component system, which is convenient for finding periodic and/or almost periodic solutions to the hierarchy of the KdV equations. It conserves the spectral functions which determine the spectrum of the auxiliary Schrödinger equation containing the solutions of the Korteweg-de Vries equations as potentials. By means of the Darboux and Abraham-Moses transformations we derive new solutions of the KdV hierarchy, which can be grasped as solitons on the fluctuating background.Some parts of the paper were delivered in the talk at the III Potsdam-V Kiev international workshop on nonlinear processes in physics, Potsdam (USA), 1–11 August, 1991.     

New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper,the generalised two-dimensional differential transform method (DTM) of solving the time-fractional coupled KdV equations is proposed.The fractional derivative is described in the Caputo sense.The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial.An illustrative example shows that the generalised two-dimensional DTM is effective for the coupled equations.     

The integrability of an extended fifth-order KdV equation with Riccati-type pseudopotential

The extended fifth-order KdV equation in fluids is investigated in this paper. Based on the concept of pseudopotential, a direct and unifying Riccati-type pseudopotential approach is employed to achieve Lax pair and singularity manifold equation of this equation. Moreover, this equation is classified into three categories: extended Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, extended Lax equation and extended Kaup–Kuperschmidt (KK) equation. The corresponding singularity manifold equations and auto-Bäcklund transformations of these three equations are also obtained. Furthermore, the infinitely many conservation laws of the extended Lax equation are found using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas.     

Hirota-Satsuma dynamics as a non-relativistic limit of KdV equations

We consider a system of two coupled KdV equations (one for left-movers, the other for right-movers) and investigate its ultra-relativistic and non-relativistic limits in the sense of BMS₃/GCA₂ symmetry. We show that there is no local ultra-relativistic limit of the system with positive energy, regardless of the coupling constants in the original relativistic Hamiltonian. By contrast, local non-relativistic limits with positive energy exist, provided there is a non-zero coupling between left- and right-movers. In these limits, the wave equations reduce to Hirota-Satsuma dynamics (of type iv) and become integrable. This is thus a situation where input from high-energy physics contributes to nonlinear science — in this case, uncovering the limiting relation between integrable structures of KdV and Hirota-Satsuma.