arXiv subjects
Publications and source records attributed to Jen-Hsu Chang.
The totally non-negative pfaffian (TNNP) is define for a skew-symmetric matrix such that all the sub-pfaffians are non-negative. It appears in the pfaffian structure of $τ$-function for the non-singular web solitons of the BKP equation . One constructs TNNP using the Perfect matching, chord diagram and the Dyck paths enumeration.Its tridiagonal form is also investigated.
One constructs the parity-time symmetric solitons in the complex KP Equation using the totally non-negative Grassmannian. We obtain that every element in the totally non-negative orthogonal Grassmannian corresponds to a parity-time symmetric solitons solution.
The solitons solution of BKP equation can be constructed by the Pfaffian structure. Then one investigates the real line solitons structure of BKP equation using the totally non-negative Grassmannian. Especially, the N-soliton solution is studied and its self-dual Tau function is obtained. Also, one can construct the totally non-negative Grassmannian of the Sawada-Kotera equation for its real line solitons.
We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions by perturbation theory, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of Hermite-Gaussian functions, but also unveil a family of peakon-like solutions without linear counterparts. As nonlinear Schr{ö}dinger wave equation has served as an important model equation in various sub-fields in physics, our proposed generalized quantum harmonic oscillator opens an unexplored area for quantum particles with nonlinear effective masses.
Soliton solutions are studied for paraxial wave propagation with intensity-dependent dispersion. Although the corresponding Lagrangian density has a singularity, analytical solutions, derived by the pseudo-potential method and the corresponding phase diagram, exhibit one- and two-humped solitons with almost perfect agreement to numerical solutions. The results obtained in this work reveal a hitherto unexplored area of soliton physics associated with nonlinear corrections to wave dispersion.
Using the Wronskian representation of $τ$-function, one can investigate the resonant structure of kink-soliton and line-soliton of the modified KP equation. It is found that the resonant structure of the the soliton graph is obtained by superimposition of the two corresponding soliton graphs of the two Le-Diagrams given an irreducible Schubert cell in a totally non-negative Grassmannian $Gr(N,M)_{ \geq 0}$. Several examples are given.
We study soliton interaction in the Modified Kadomtsev-Petviashvili-(II) equation (MKP-(II)) using the totally non-negative Grassmannian. One constructs the multi-kink soliton of MKP equation using the $τ$-function and the Binet-Cauchy formula, and then investigates the interaction between kink solitons and line solitons. Especially, Y-type kink-soliton resonance, O-type kink soliton and P-type kink soliton of X-shape are investigated. Their amplitudes of interaction are computed after choosing appropriate phases.
Inspired by the works of Y. Ohta and J. Yang, one constructs the lumps solutions in the Kadomtsev-Petviashvili-(I) equation using the Grammian determinants. It is shown that the locations of peaks will depend on the real roots of Wronskian of the orthogonal polynomials for the asymptotic behaviors in some particular cases. Also, one can prove that all the locations of peaks are on a vertical line when time approaches - $\infty$, and then they will be on a horizontal line when time approaches $\infty$, i.e., there is a rotation $\fracπ{2}$ after interaction.
Using the reality condition of the solutions, one constructs the Mach-type soliton of the Novikov-Veselov equation by the minor-summation formula of the Pfaffian. We study the evolution of the Mach-type soliton and find that the amplitude of the Mach stem wave is less than two times of the one of the incident wave. It is shown that the length of the Mach stem wave is linear with time. One discusses the relations with V -shape initial value wave for different critical values of Miles parameter.
Using the reality condition of the solutions, one constructs the real Pfaffian N-solitons solutions of the Novikov-Veselov (NV) equation using the $\tan$ function and the Schur identity. By the minor-summation formula of the Pfaffian, we can study the interactions of solitons in the Novikov-Veselov equation from the Kadomtsev-Petviashvili (KP) equation's point of view, that is, the totally non-negative Grassmannian. Especially, the Y-shape resonance, O-type and the P-type interactions of X-shape are investigated. Also, the maximum amplitude of the intersection of the line solitons and the critical angle are computed and one makes a comparison with the KP-(II) equation.
We construct the $N$-solitons solution in the Novikov-Veselov equation from the extended Moutard transformation and the Pfaffian structure. Also, the corresponding wave functions are obtained explicitly. As a result, the property characterizing the $N$-solitons wave function is proved using the Pfaffian expansion. This property corresponding to the discrete scattering data for $N$-solitons solution is obtained in [arXiv:0912.2155] from the $\bar\partial$-dressing method.
We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov (NV) equation. The root dynamics of the $σ$-flow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particulr, when $N=3,4,5$, one can get the Gold-fish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.
We study Manakov-Santini equation, starting from Lax-Sato form of associated hierarchy. The waterbag reduction for Manakov-Santini hierarchy is introduced. Equations of reduced hierarchy are derived.We construct new coordinates transforming non-hydrodynamic evolution of waterbag reduction to non-homogeneous Riemann invariants form of hydrodynamic type.
We investigate the integrable $(2+1)$-dimensional generalized dispersionless KP (GdKP) equation (or Manakov-Santinit system) from the Lax-Sato form. Several particular three-component reductions are considered so that the GdKP equation can be reduced to hydrodynamic systems. Then one can construct infinite exact solutions of GdKP by the generalized hodograph method.
We investigate the dispersionless Veselov-Novikov (dVN) equation based on the framework of dispersionless two-component BKP hierarchy. Symmetry constraints for real dVN system are considered. It is shown that under symmetry reductions, the conserved densities are therefore related to the associated Faber polynomials and can be solved recursively. Moreover, the method of hodograph transformation as well as the expressions of Faber polynomials are used to find exact real solutions of the dVN hierarchy.
We construct the free energy associated with the waterbag model of dToda. Also, the relations of conserved densities are investigated
We construct the bi-Hamiltonian structure of the waterbag model of dKP and establish the third-order Hamiltonian operator associated with the waterbag model. Also, the symmetries and conserved densities of rational type are discussed.
We investigate the bi-Hamiltonian structure of the waterbag model of dKP for two component case. One can establish the third-order and first-order Hamiltonian operator associated with the waterbag model. Also, the dispersive corrections are discussed.