Narrow Results

The rational solitons and other wave solutions demonstrate the existence of wave packets propagating through optical Kerr media, exhibiting non-diffractive and non-dispersive spatiotemporal localization. This paper focuses on constructing rational solitons, multi-wave solutions, and other solutions for a dimensionless time-dependent paraxial wave model (dt-pPWM). These solutions are derived through the application of symbolic computation and the ansatz function method, using both traveling wave and logarithmic transformations. Our approach encompasses three distinct methods: the positive quadratic function approach, the three-wave approach, and the double exponential approach. These wave results find practical applications and contribute to a deeper understanding of the physical phenomena described by this model. We have computed conservative quantities corresponding to the power, momentum, and energy of solitons. To assess the stability of the wave equation, we conducted modulational instability analysis, confirming the stability and accuracy of all soliton solutions. The structure of the solution provides a physical interpretation. By selecting appropriate parametric values, we generated 3D graphics illustrating various multi-wave solutions, including lump waves, rogue waves, and multipeak solitons. Additionally, we observed intriguing phenomena arising from the interaction of multi-waves, which have applications in telecommunications, quantum optics and other related fields.

We develop the theory of the momentum map for the Maxwell–Lorentz equations for a spinning extended charged particle. The development relies on the Hamilton–Poisson structure of the system. This theory is indispensable for the study of long-time behavior and radiation of the solitons of this system, in particular for the proof of orbital stability of the corresponding solitons moving with constant speed and rotating with constant angular velocity.

We apply the theory to the translation and rotation symmetry groups of the system. The general theory of the momentum map results in formal equations for the corresponding invariants. We solve these equations obtaining expressions for the momentum and the angular momentum.

We check the conservation of these expressions and their coincidence with known classical invariants. For the first time, we specify a general class of finite energy initial states which provide the conservation of angular momentum.

We consider two kinds of stability (under a perturbation) of the ground state of a self-adjoint operator: the one is concerned with the sector to which the ground state belongs and the other is about the uniqueness of the ground state. As an application to the Wigner–Weisskopf model which describes one mode fermion coupled to a quantum scalar field, we prove in the massive case the following: (a) For a value of the coupling constant, the Wigner–Weisskopf model has degenerate ground states; (b) for a value of the coupling constant, the Wigner–Weisskopf model has a first excited state with energy level below the bottom of the essential spectrum. These phenomena are nonperturbative.

Though a global Chern–Simons (2k-1)-form is not gauge-invariant, this form seen as a Lagrangian of higher-dimensional gauge theory leads to the conservation law of a modified Noether current.

Recently, a 4-index generalization of the Einstein theory has been proposed by Moulin [F. Moulin, Eur. Phys. J. C77, 878 (2017)]. Using this method, we find the most general 2-index field equations derivable from the Einstein–Hilbert action. The application of Newtonian limit, the role of gravitational coupling constant and the effects of the properties of ordinary energy–momentum tensor in obtaining a 4-index gravity theory have been studied. We also address the results of building Weyl free 4-index gravity theory. Our study displays that both the Einstein and Rastall theories can be obtained as the subclasses of a 4-index gravity theory which shows the power of 4-index method in unifying various gravitational theories. It is also obtained that the violation of the energy–momentum conservation law may be allowed in 4-index gravity theory, and moreover, the contraction of 4-index theory generally admits a non-minimal coupling between geometry and matter field in the Rastall way. This study also shows that, unlike the Einstein case, the gravitational coupling constant of 4-index Rastall theory generally differs from that of the ordinary 2-index Rastall theory.

We construct a classical dynamical system whose phase space is a certain infinite-dimensional Grassmannian manifold, and propose that it is equivalent to the large N limit of two-dimensional QCD with an O(2N+1) gauge group. In this theory, we find that baryon number is a topological quantity that is conserved only modulo 2. We also relate this theory to the master field approach to matrix models.

An algorithm of constructing the polynomial type conservation laws of nonlinear evolution equations (either parameterized or non-parameterized) is described and a Maple package to automate the computation is presented. For parameterized equation(s), can determine the parameters constraints automatically such that a sequence of conservation laws exist.

A family of new discrete nonlinear equations associated with the Lotka–Volterra lattice is constructed by defining a novel algebraic system X. Moreover, some properties of the hierarchy are discussed. Furthermore, nonlinear integrable couplings of the resulting hierarchy are derived from an extended algebraic system . Finally, an example is given.

A discrete three-by-three matrix spectral problem is put forward and the corresponding discrete soliton equations are deduced. By means of the trace identity the Hamiltonian structures of the resulting equations are constructed, and furthermore, infinitely many conservation laws of the corresponding lattice system are obtained by a direct way.

A systematic and unified method is presented for the derivation of the conserved quantities for the laminar classical wake and the wake of a self-propelled body. The multiplier method for the derivation of conservation laws is applied to the far downstream wake equations expressed in terms of the velocity components which gives rise to a second-order system of two partial differential equations, and in terms of the stream function which results in one third-order partial differential equation. Once the corresponding conservation laws are obtained, they are integrated across the wake and upon imposing the boundary conditions and the condition that the drag on a self-propelled body is zero, the conserved quantities for the classical wake and the wake of a self-propelled body are derived. In addition, this method results in the discovery of a new laminar wake which may have physical significance.

In this paper, we study a 6-field integrable lattice system, which, in some special cases, can be reduced to the self-dual network equation, the discrete second-order nonlinear Schrödinger equation and the relativistic Volterra lattice equation. With the help of the Lax pair, we construct infinitely many conservation laws and a new Darboux transformation for system. Exact solutions resulting from the obtained Darboux transformation are presented by using a given seed solution. Further, we generate the soliton solutions and plot the figures of one-soliton solutions with properly parameters.

In this paper, a $(2+1)$-dimensional modified Heisenberg ferromagnetic system, which appears in the biological pattern formation and in the motion of magnetization vector of the isotropic ferromagnet, is being investigated with the aim of exploring its similarity solutions. With the aid of Lie symmetry analysis, this system of partial differential equations has been reduced to a new system of ordinary differential equations, which brings an analytical solution of the main system. Infinitesimal generators, commutator table, and the group-invariant solutions have been carried out by using Lie symmetry approach. Moreover, conservation laws of the above mentioned system have been obtained by utilizing the new conservation theorem proposed by Ibragimov. By applying this analysis, the obtained results might be helpful to understand the physical structure of this model and show the authenticity and effectiveness of the proposed method.

A new Lax pair is introduced for a perturbed Kaup–Newell equation and used to construct two series of conservation laws through a Riccati equation that a ratio of eigenfunctions satisfies. Both series of conservation laws are defined recursively, and the first two in each series are presented explicitly.

For describing the propagation of ultrashort pulses in the birefringent or two-mode fiber, we consider the coupled nonlinear Schrödinger equations with higher-order effects. According to the Ablowitz–Kaup–Newell–Segur system, 3$\times$3 Lax pair for such equations is derived. Based on the Lax pair, we construct the conservation laws and Darboux transformation (DT). One- and two-soliton solutions are obtained via the DT, and we graphically present the one soliton and bound-state two solitons.

Under investigation with symbolic computation in this paper, is a variable-coefficient Sasa–Satsuma equation (SSE) which can describe the ultra short pulses in optical fiber communications and propagation of deep ocean waves. By virtue of the extended Ablowitz–Kaup–Newell–Segur system, Lax pair for the model is directly constructed. Based on the obtained Lax pair, an auto-Bäcklund transformation is provided, then the explicit one-soliton solution is obtained. Meanwhile, an infinite number of conservation laws in explicit recursion forms are derived to indicate its integrability in the Liouville sense. Furthermore, exact explicit rogue wave (RW) solution is presented by use of a Darboux transformation. In addition to the double-peak structure and an analog of the Peregrine soliton, the RW can exhibit graphically an intriguing twisted rogue-wave (TRW) pair that involve four well-defined zero-amplitude points.

A new coupled Burgers equation and a new coupled KdV equation which are associated with $3\times 3$ matrix spectial problem are investigated for complete integrability and covariant property. For integrability, Lax pair and conservation laws of the two new coupled equations with four potentials are established. For covariant property, Darboux transformation (DT) is used to construct explicit solutions of the two new coupled equations.

In this paper, we firstly establish infinitely many conservation laws of the 3-coupled integrable lattice equations by using the Riccati method. Comparing with the results obtained by Sahadevan and Balakrishnan, we not only get infinite conserved densities of the polynomial form, but also some conserved densities of logarithmic form. Secondly, Darboux transformation for the system is derived with the help of the Lax pair and gauge transformation. Finally, we obtain the exact solutions of the system with the obtained Darboux transformation, and present the soliton solutions and their figures with properly parameters.

A recently defined (3+1)-dimensional extended quantum Zakharov–Kuznetsov (QZK) equation is examined here by using the Lie symmetry approach. The Lie symmetry analysis has been used to obtain the varieties in invariant solutions of the extended Zakharov–Kuznetsov equation. Due to existence of arbitrary functions and constants, these solutions provide a rich physical structure. In this paper, the Lie point symmetries, geometric vector field, commutative table, symmetry groups of Lie algebra have been derived by using the Lie symmetry approach. The simplest equation method has been presented for obtaining the exact solution of some reduced transform equations. Finally, by invoking the new conservation theorem developed by Nail H. Ibragimov, the conservation laws of QZK equation have been derived.

In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.

Starting from a more generalized discrete $2\times 2$ matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the $N$-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete $N$-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.