Short Pulse Equation Research Articles

The exploding solitons of the sine–Gordon equation

In this paper, the hodograph equivalent short pulse (HESP) equations are investigated via the Darboux transformation, we derive the soliton and positon solutions from the “seed” solutions, and then, the decomposition of the lower-order positons into single-solitons is given analytically when time T is sufficiently large. As a notable new result, we obtain the exploding soliton and positon solutions of the sine–Gordon (SG) equation from the hodograph equivalent short pulse equations.

Breathers and rogue waves on the plane wave/periodic background for the generalized complex short pulse equation

Breathers and rogue waves on the plane wave/periodic background for the generalized complex short pulse equation

A Riemann–Hilbert approach to solution of the modified focusing complex short pulse equation

A Riemann–Hilbert approach to solution of the modified focusing complex short pulse equation

Soliton resolution for the modified short pulse equation with the weighted Sobolev initial data

In this paper, the Cauchy problem of the modified short pulse (mSP) equation with initial conditions in weighted Sobolev space is studied by using [Formula: see text]-steepest descent method. Based on the spectral analysis of Lax pair and the transformations of field variables and independent variables, a basic Riemann–Hilbert problem (RHP) is established, and the solution of Cauchy problem for the mSP equation is transformed into the corresponding RHP. The asymptotic expansion of the solution of mSP equation is derived in a fixed space-time cone [Formula: see text] According to this, the soliton resolution conjecture of the mSP equation is given, in which the leading term is characterized by an [Formula: see text]-soliton on the discrete spectrum, the second term comes from the soliton–radiation interactions on the continuum spectrum, and the error term [Formula: see text] is generated by the corresponding [Formula: see text]-problem.

Multibreather-like solutions of the real and complex reverse space–time nonlocal defocusing short-pulse equations

Multibreather-like solutions of the real and complex reverse space–time nonlocal defocusing short-pulse equations

Lax pairs informed neural networks solving integrable systems

Lax pairs are one of the most important features of integrable system. In this work, we propose the Lax pairs informed neural networks (LPNNs) tailored for the integrable systems with Lax pairs by designing novel network architectures and loss functions, comprising LPNN-v1 and LPNN-v2. The most noteworthy advantage of LPNN-v1 is that it can transform the solving of complex integrable systems into the solving of relatively simple Lax pairs, and it not only efficiently solves data-driven localized wave solutions, but also obtains spectral parameter and corresponding spectral function in spectral problems of the integrable systems. On the basis of LPNN-v1, we additionally incorporate the compatibility condition/zero curvature equation of Lax pairs in LPNN-v2, its major advantage is the ability to solve and explore high-accuracy data-driven localized wave solutions and associated spectral problems for integrable systems with Lax pairs. The numerical experiments focus on studying abundant localized wave solutions for very important and representative integrable systems with Lax pairs, including the soliton solution of the Korteweg-de Vries (KdV) equation and modified KdV equation, rogue wave solution of the nonlinear Schrödinger equation, kink solution of the sine-Gordon equation, non-smooth peakon solution of the Camassa-Holm equation and pulse solution of the short pulse equation, as well as the line-soliton solution of Kadomtsev-Petviashvili equation and lump solution of high-dimensional KdV equation. The innovation of this work lies in the pioneering integration of Lax pairs informed of integrable systems into deep neural networks, thereby presenting a fresh methodology and pathway for investigating data-driven localized wave solutions and spectral problems.

The higher-order positon and breather-positon solutions for the complex short pulse equation

The higher-order positon and breather-positon solutions for the complex short pulse equation

WITHDRAWN: Dynamics of solitons for the sine–Gordon equation

In this paper, the hodograph equivalent short pulse equations are investigated via the Darboux transformation, we derive the soliton and positon solutions from the “seed” solution, and then, the decomposition of the lower-order positons into single-solitons is given analytically when time T is sufficiently large. As a notable new result, we obtain the exploding soliton and positon solutions of the sine-Gordon equation from the hodograph equivalent short pulse equations.

Multiloop soliton solutions and compound WKI–SP hierarchy

AbstractIn this paper, a compound equation which is a mix of the Wadati–Konno–Ichikawa (WKI) equation and the short‐pulse (SP) equation is first studied. By transforming both the independent and dependent variables in the equation, we introduce a novel hodograph transformation to convert the compound WKI–SP equation into the mKdV–SG (modified Korteweg–de Vries and sine‐Gordon) equation. The multiloop soliton solutions in the form of the parametric representation are found. It is shown that the ‐loop soliton solution may be decomposed exactly into separate soliton elements by using a Moloney–Hodnett‐type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of and are investigated in detail. The corresponding phase shifts of each loop or antiloop soliton caused by its interaction with the other ones are calculated. Furthermore, a new hierarchy of WKI–SP‐type equations possessing multiloop soliton solutions is constructed. These deduced equations are all with time‐varying coefficients and the corresponding dispersion relation will have a time‐dependent velocity. The whole hierarchy of equations which include the WKI‐type equations, the SP‐type equations, and the compound generalized WKI–SP equations, are illustrated Lax integrable. The specific equation in the hierarchy is labeled as equation so that its Lax pairs can be directly written out with the help of and . A unified hodograph transformation is established to relate the compound WKI–SP hierarchy with the mKdV–SG hierarchy.

Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data

We characterize the long-time asymptotic behavior of the solution of the initial value problem for the coupled complex short-pulse equation associated with the 4×4 matrix spectral problem. The spectral analysis of the 4×4 matrix spectral problem is very difficult because of the existence of energy-dependent potentials and the WKI type. The method we adopted is a combination of the inverse scattering transform and Deift-Zhou nonlinear steepest descent method. Starting from the Lax pair associated with the coupled complex short-pulse equation, we derive a basic Riemann-Hilbert problem by introducing some appropriate spectral function transformations, and reconstruct the potential parameterized from the solution of the basic Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at k→0. We finally obtain the leading order asymptotic behavior of the solution of the coupled complex short-pulse equation through a series of Deift-Zhou contour deformations.

Long-time asymptotics for the coupled modified complex short-pulse equation

Long-time asymptotics for the coupled modified complex short-pulse equation

Inverse scattering transform for the coupled modified complex short pulse equation: Riemann–Hilbert approach and soliton solutions

In this paper, we develop a Riemann–Hilbert (RH) approach to the inverse scattering transform for the coupled modified complex short pulse (cmcSP) equation, which appeared as a reduction of the four-component system of short pulse type introduced by Popowicz in 2017. This approach allows us to present the general soliton solutions of the cmcSP equation in parametric form. Compared with the early results for the scalar modified complex short pulse equation, the cmcSP equation possesses richer soliton solutions, which include smooth solitons, cuspons, breathers and their various interactions.

Rogue periodic waves of the short pulse equation and the coupled integrable dispersionless equation

In this paper, we explore the topic of rogue waves on periodic backgrounds. Using the travelling wave reduction method, we derive two families of periodic solutions for the coupled integrable dispersionless (CD) equation expressed in Jacobi elliptic dnoidal and conoidal functions, respectively. We use these Jacobi elliptic functions as seed solutions to construct algebraic decaying solitons on the dnoidal background of the CD equation based on the one-fold Darboux transformation (DT). Furthermore, we utilize the two-fold DT to find rogue wave solutions on the conoidal background of the CD equation. To obtain solutions of the short pulse equation, we use the hodograph transformation. We analyse the dynamics of the explicit solutions obtained from this research through plots.

Spectral stability of elliptic solutions to the short-pulse equation

The short pulse (SP) equation describes the propagation of ultrashort optical pulses in nonlinear media and possesses a Lax pair of the Wadati–Konno–Ichikawa (WKI) type. In this paper, using integrability, we examine the spectral stability of elliptic solutions to the SP equation. Firstly, we analytically give an explicit description of the spectrum of the WKI-Lax operator to the SP equation for elliptic potentials. Then, by constructing the squared-eigenfunction connection between the non-standard linear stability problem (LZ=ΛZ′) and the Lax spectral problem, we prove that the elliptic solutions are spectrally stable with respect to subharmonic perturbations.

On the asymptotic stability of N-soliton solution for the short pulse equation with weighted Sobolev initial data

In this work, we are devoted to studying the Cauchy problem of the short pulse (SP) equation with weighted Sobolev initial data. By developing the ∂¯-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q(x,t) in solitonic region of space-time, (yt)=ξ for any fixed ξ=ξ0∈(0,+∞), and give bounds for the error decaying as |t|→∞. Based on the resulting asymptotic behavior, the asymptotic approximation of the SP equation is characterized with the soliton term confirmed by N(I)-soliton on discrete spectrum with residual error up to O(t−1). Combining with previous results presented by Yang and Fan, that is the long time asymptotic expansion of the solution q(x,t) in solitonic region ξ=yt∈(−∞,0), our results show that the soliton resolution conjecture of the SP equation is fully solved in all space-time solitonic regions. Moreover, considering properties of the scattering map of SP equation, we derive an asymptotic stability result for a class of initial values.

Multi-soliton solutions for the two-component modified short pulse equation and its nonlocal reductions via Bäcklund transformations

Multi-soliton solutions for the two-component modified short pulse equation and its nonlocal reductions via Bäcklund transformations

Interaction Behaviors Between Solitons, Breathers and Their Hybrid Forms for a Short Pulse Equation

Interaction Behaviors Between Solitons, Breathers and Their Hybrid Forms for a Short Pulse Equation

Negative order mKdV equation and [formula omitted]-solitons

In this paper, we investigate a negative order modified Korteweg–de Vries equation (nmKdV). Utilizing Hirota’s bilinear method, Bäcklund transformations are constructed to obtain explicit solutions for N-solitons. The solitons exhibit cuspon-like behavior under certain parameter selections. Furthermore, connections between the nmKdV equation and other soliton equations, such as the short pulse equation, the sine-Gordon equation, and the Liouville equation, are investigated.

Periodic solutions of the modified short pulse equation

Periodic solutions of the modified short pulse equation

Loop dynamics of a fully discrete short pulse equation

In this paper, a fully discrete short pulse (SP) equation is presented as an integrability condition of a linear system of difference equations (also known as discrete Lax pair). Additionally, two semi-discrete versions of the SP equation have also been obtained from fully discrete SP equation under continuum limits. Darboux transformation is employed to compute multi-soliton solutions of fully discrete and semi-discrete SP equations. Explicit expressions of first and second nontrivial soliton solutions are computed. We also derived explicit expression of breather solution for fully discrete SP equation. The dynamics of single loop soliton and interaction mechanism of loop–loop and loop–antiloop solutions have been explored and illustrated.