The explicit symmetry breaking solutions of the Kadomtsev-Petviashvili equation

To describe two correlated events, the Alice–Bob (AB) systems were constructed by Lou through the symmetry of the shifted parity, time reversal and charge conjugation. In this paper, the coupled AB system of the Kadomtsev–Petviashvili equation, which is a useful model in natural science, is established. By introducing an extended Bäcklund transformation and its bilinear formation, the symmetry breaking soliton, lump and breather solutions of this system are derived with the aid of some ansatze functions. Figures show these fascinating symmetry breaking structures of the explicit solutions.     

Branching equation of Andronov-Hopf bifurcation under group symmetry conditions

In branching theory of solutions of nonlinear equations group analysis methods [Ovsyannikov, Group Analysis of Differential Equations (Nauka, Moscow, 1978); Lectures on the Theory of Group Properties of Differential Equations (Novosibirsk University, Novosibirsk, 1966)] give the general approach for the construction of the complete form of branching equation and its subsequent investigation. These methods are applied here to the general situation of Andronov-Hopf bifurcation when there are some multiple semisimple eigenvalues on imaginary axis. (c) 1997 American Institute of Physics.     

Stochastic Kadomtsev-Petviashvili equation

The example of Kadomtsev-Petviashvili equations with a random time-dependent force (stochastic Kadomtsev-Petviashvili equations) is used to show that the theory of Brownian particle motion can be applied to the theory of the stochastic behavior of solitons of model hydrodynamic equations which are completely integrable in the absence of forces and interrelated by the generalized Galilean transformation. The Brownian motion of two-dimensional algebraic solitons of the Kadomtsev-Petviashvili equations with positive dispersion leads to their diffusion broadening similar to the broadening of one-dimensional solitons of other fully integrable hydrodynamic equations. However, for longer times the rate of decay of algebraic solitons is higher because of the degeneracy of the momentum integral for these solitons. The behavior of a periodic chain of algebraic solitons is established under the action of a random force. Tilted plane solitons of the Kadomtsev-Petviashvili equations with negative dispersion vary under the action of a random force similar to the solitons of the Korteweg-de Vries equation. Several of these solitons interact via “virtual solitons” and generate new solitons provided that resonance conditions are satisfied whose dimensions increase as a result of the influence of the random force.     

Two integrable extensions of the Kadomtsev-Petviashvili equation

In this work, two new completely integrable extensions of the Kadomtsev-Petviashvili (eKP) equation are developed. Multiple soliton solutions and multiple singular soliton solutions are derived to demonstrate the compatibility of the extensions of the KP equation.     

Line soliton interactions of the Kadomtsev-Petviashvili equation

We study soliton solutions of the Kadomtsev-Petviashvili II equation (-4u(t)+6uu(x)+3u(xxx))(x)+u(yy)=0 in terms of the amplitudes and directions of the interacting solitons. In particular, we classify elastic N-soliton solutions, namely, solutions for which the number, directions, and amplitudes of the N asymptotic line solitons as y-->infinity coincide with those of the N asymptotic line solitons as y-->-infinity. We also show that the (2N-1)!! types of solutions are uniquely characterized in terms of the individual soliton parameters, and we calculate the soliton position shifts arising from the interactions.     

Generalized renormalization group equation for systems of arbitrary symmetry

An exact renormalization group equation for the Ginzburg-Landau-Wilson functional of an arbitrary symmetry is obtained. The equation derived does not contain redundant operators which must be transformed away.     

The Kadomtsev-Petviashvili II equation on the half-plane

The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics, where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey-Stewartson equation (Fokas (2009) [13]), in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d-bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial value problems in 2+1 is that the d-bar formalism now involves a function in the complex plane which is discontinuous across the real axis.     

On Ground-Traveling Waves for the Generalized Kadomtsev-Petviashvili Equations

As a continuation of our previous work, we improve some results on convergence of periodic KP traveling waves to solitary ones as the period goes to infinity. In addition, we present some qualitative properties of such waves, as well as nonexistence results, in the case of general nonlinearities. We suggest an approach which does not use any scaling argument. This revised version was published online in July 2006 with corrections to the Cover Date.     

Painlevé analysis and symmetry group for the coupled Zakharov-Kuznetsov equation

The Painlevé property for the coupled Zakharov-Kuznetsov equation is verified with the WTC approach and new exact solutions of bell-type are constructed from standard truncated expansion. A symmetry transformation group theorem is also given out from a simple direct method.     

Separation of variables and symmetry operators for the conformally invariant Klein-Gordon equation on curved spacetime

The separability of the conformally invariant Klein-Gordon equation and the Laplace-Beltrami equation are contrasted on two classes of Petrov type D curved spacetimes, showing that neither implies the other. The second-order symmetry operators corresponding to the separation of variables of the conformally invariant Klein-Gordon equation are constructed in both classes and the most general second-order symmetry operator for the conformally invariant Klein-Gordon operator on a general curved background is characterized tensorially in terms of a valence two-symmetric tensor satisfying the conformal Killing tensor equation and further constraints.     

Finite symmetry transformation group of the Konopelchenko-Dubrovsky equation from its Lax pair

Starting from a weak Lax pair,the general Lie point symmetry group of the Konopelchenko-Dubrovsky equation is obtained by using the general direct method.And the corresponding Lie algebra structure is proved to be a Kac-Moody-Virasoro type.Furthermore,a new multi-soliton solution for the Konopelchenko-Dubrovsky equation is also given from this symmetry group and a known solution.     

Modified Kadomtsev-Petviashvili equation in cold collisionless plasma

Nonlinear electromagnetic wave propagation through cold collisionless plasma in (2+1) dimensions is studied using the nonlinear reductive perturbation method. It is shown that to the lowest order of perturbation, the system of equations can be reduced to modified Kadomtsev-Petviashvili equation.     

Finite symmetry transformation group of the Konopelchenkoben Dubrovsky equation from its Lax pair

Starting from a weak Lax pair, the general Lie point symmetry group of the Konopelchenko-Dubrovsky equation is obtained by using the general direct method. And the corresponding Lie algebra structure is proved to be a Kac-Moody-Virasoro type. Furthermore, a new multi-soliton solution for the Konopelchenko-Dubrovsky equation is also given from this symmetry group and a known solution.     

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

We construct the formal solution to the Cauchy problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrödinger operator for a quantum particle in a time-dependent potential.     

On a new hierarchy of symmetries for the Kadomtsev-Petviashvili equation

We present a new hierarchy of symmetries for the Kadomtsev-Petviashvili equation. These new symmetries depend on the space and time variables explicitly. Together with the previously known classical symmetries, they constitute an infinite-dimensional Lie algebra.     

Grammian solutions and Pfaffianization of the non-isospectral modified Kadomtsev-Petviashvili equation

The Grammian determinant solutions of the non-isospectral modified Kadomtsev-Petviashvili (mKP) equation are presented. Moreover, a new non-isospectral coupled system is constructed by using the Pfaffianization procedure. Furthermore, Gramm-type Pfaffian solutions of the non-isospectral coupled system are obtained.     

Dirac's equation and a new group of external symmetry

It is shown how the handedness of the massless leptons, and the Poincaré invariance of leptonic interactions, can be based on a new group of external symmetry.     

Callan-Symanzik and renormalization group equation in a model with spontaneous breaking of the symmetry

In a simple model with spontaneous breaking of the axialU(1)-symmetry via the Higgs mechanism we construct the Callan-Symanzik and renormalization group equation in the Goldstone mode. Aiming at questions of renormalization group improvement and the like we compare two different parametrizations the model can be described with. We show that in the presence of fermions a β-function for a physical mass or some equivalent of it enters unavoidably the Callan-Symanzik equation, which leads to significant differences to the symmetric theory starting with two loops. On the other hand in the asymptotic region the equivalence to the symmetric theory is manifest.     

The double Wronskian solutions of a non-isospectral Kadomtsev-Petviashvili equation

The double Wronskian solutions of a non-isospectral Kadomtsev-Petviashvili equation (n-KPE) are derived. One-soliton solution and two-soliton solution are presented, the characteristics of one-soliton and two-soliton scattering are discussed also.