Nonlocal symmetry and interaction solutions for the new (3+1)-dimensional integrable Boussinesq equation

The nonlocal symmetry of the new (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlev\'{e} method. The nonlocal symmetry can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables. The finite symmetry transformation related to the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is studied. Meanwhile, the new (3+1)-dimensional Boussinesq equation is proved by the consistent tanh expansion method and many interaction solutions among solitons and other types of nonlinear excitations such as cnoidal periodic waves and resonant soliton solution are given.

Multi-soliton dynamics of anti-self-dual gauge fields

We study dynamics of multi-soliton solutions of anti-self-dual Yang-Mills equations for G = GL(2, ℂ) in four-dimensional spaces. The one-soliton solution can be interpreted as a codimension-one soliton in four-dimensional spaces because the principal peak of action density localizes on a three-dimensional hyperplane. We call it the soliton wall. We prove that in the asymptotic region, the n-soliton solution possesses n isolated localized lumps of action density, and interpret it as n intersecting soliton walls. More precisely, each action density lump is essentially the same as a soliton wall because it preserves its shape and “velocity” except for a position shift of principal peak in the scattering process. The position shift results from the nonlinear interactions of the multi-solitons and is called the phase shift. We calculate the phase shift factors explicitly and find that the action densities can be real-valued in three kind of signatures. Finally, we show that the gauge group can be G = SU(2) in the Ultrahyperbolic space 𝕌 (the split signature (+, +, −, −)). This implies that the intersecting soliton walls could be realized in all region in N=2 string theories. It is remarkable that quasideterminants dramatically simplify the calculations and proofs.

Removing tadpoles in a soliton sector

It has long been known that perturbative calculations can be performed in a soliton sector of a quantum field theory by using a soliton Hamiltonian, which is constructed from the defining Hamiltonian by shifting the field by the classical soliton solution. It is also known that even if tadpoles are eliminated in the vacuum sector, they remain in the soliton sector. In this note we show, in the case of quantum kinks at two loops, that the soliton sector tadpoles may be removed by adding certain quantum corrections to the classical solution used in this construction. Stated differently, the renormalization condition that the soliton sector tadpoles vanish may be satisfied by renormalizing the soliton solution.

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.

Lumps and interaction solutions to the (4 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

In this paper, we introduce a new nonlinear evolution equation, which is ([Formula: see text])-dimensional variable-coefficient Kadomtsev–Petviashvili equation. First, according to the Hirota bilinear method, we get some exact solutions of the equation, including lump solution, lump-soliton solution, rogue-soliton solution and lump-kink solution. Then, we obtain some new exact solutions by generalizing the form of the lump solution on a further solution. Finally, based on the symbolic calculation method with Mathematica, the characteristics of the interaction solutions are shown in the graphs and we analyze the dynamic change of the solutions. Furthermore, we discuss the applications of these solutions in physics via the analysis.

Riemann-Hilbert Approach of The Complex Sharma-Tasso-Olver Equation and Its N-Soliton Solutions

In this paper, we use Riemann-Hilbert method to study the N-soliton solutions of the complex Sharma-Tasso-Olver(cSTO) equation. And then, based on analyzing the spectral problem of the Lax pair, the matrix Riemann-Hilbert problem for this integrable equation can be constructed, the N-soliton solutions about this system are given explicitly under the relationship of scattering matrix. At last, under the condition that some specifific parameter values are given, the three-dimensional diagram of the 2-soliton solution and the trajectory of the soliton solution will be simulated.