Global solution to the complex short pulse equation

1.   Introduction

The short pulse equation

was derived by Schäfer and Wayne ^[1] as a nonlinear model to describe the propagation of ultra-short optical pulses in isotropic optical fibers. Here, $u = u(t, x)$ is a real-valued function, representing the magnitude of the electric field, and the subscripts denote partial derivatives with respect to $t$ and $x$. It is an integrable differential equation and has attracted much attention in the past two decades. Various solutions to this equation have been obtained, including, its periodic and solitary wave solutions in ^[2]; loop and pulse solutions in ^[3]; two-loop soliton solutions in ^[4]; and multiloop solutions, multibreather, and periodic solutions in ^[5,6]. Concerning the Cauchy problem of (1.1), local well-posedness of solution was obtained in ^[1,7], where in ^[7] the global existence of the solution for small initial data in $H^2$ was also established, and modified scattering behavior was proved in ^[8,9,10] under different conditions on initial data.

A general model (the generalized Ostrovsky equation) related to the short pulse equation is

where $p\geq 2$ is an integer. The case of $p = 2$ is usually referred as the Ostrovsky-Hunter equation ^[11] and short-wave equation ^[12]. For $p\geq 4$, the global well-posedness and scattering was proved by Stefanov et al. ^[13] and Hayashi et al. ^[14]. Apart from the above generalization of the short pulse equation (1.1), there are several other different versions, such as, the higher-order nonlinearity corrections in ^[15], the vector short pulse equations in ^[16,17], and the multi-component short pulse model ^[18,19].

In this paper we study the complex short pulse equation

where $q(t, x): \mathbb{R}^+\times\mathbb{R}\rightarrow \mathbb{C}$, is a complex-valued function. Equation (1.3) was produced from the negative order Wadati-Konno-Ichikawa (WKI) hierarchy in ^[20,21], where the Lax pair for the whole WKI hierarchy and algebraic structure with r-matrix were discussed. See ^[18,22] for the derivation of this equation and ^[23,24,25] for the symmetry methods to solve the equations. Zhaqilao et al. ^[19] studied multi-soliton solutions and the Cauchy problem for (1.3). As far as we know, the global existence theory for this equation has not been established, and the goal of this paper is to obtain the global well-posedness result.

To apply the partial differential equation theories for (1.3), we rewrite it into a first order equation with respect to $t$. Hence, we integrate Eq (1.3) to get

where $\partial_x^{-1}q$ is defined through the Fourier transform, namely,

From now on, we mainly focus on Eq (1.4).

Throughout the paper, $L^p$ $(p\geq 1)$ is the usual Lebesgue space, $H^k$ denotes the inhomogeneous Sobolev spaces equipped with the norm

and $\dot{H}^{-1}$ is the homogeneous Sobolev space with

The main results of the paper are the following two theorems.

Theorem 1.1. Assume that the initial data $q_0\in {H^2}\cap \dot{H}^{-1}$. Then, there exists a time $T > 0$, depending only on the norms of initial data, such that the complex short pulse equation (1.4) has a unique solution $q\in C([0, T]; H^2\cap \dot{H}^{-1})$ satisfying $q(0) = q_0$. Moreover, if $T^*$ is the time that the solution can not be continued to $T = T^*$, then either $T^* = \infty$ or $\|q(t, x)\|_{H^2}$ tends to infinite as $t\rightarrow T^*$.

We remark that the sine-Gordon transformation method studying the Eq (1.1) in ^[7] doesn't work for the complex Eq (1.3). Here, we use regularized technology to compensate the loss of derivative for the nonlinear term and take approximation argument to obtain the local existence result for the complex short pulse equation.

To state the global result, we set

and we will show that they are all conserved as long as the solution exists (see Lemma 4.1 in Section 4). We note that the second term in the quantity $\mathcal{H}_3(0)$ is new and nonzero in our complex case. The conserved quantity $\mathcal{H}_3$ is derived by delicate analysis and computations. We now state the global existence of solution to (1.3). Due to the work of Liu et al. ^[26,27] where wave-break phenomena is demonstrated, the global result obtained here requires the smallness assumption on the initial data.

Theorem 1.2. Let $q_0\in H^2\cap \dot{H}^{-1}$ with $\mathcal{H}_1(0), \mathcal{H}_2(0), \mathcal{H}_3(0)$ small. Then, the complex short pulse equation (1.3) admits a unique global solution $q\in C([0, +\infty); H^2\cap \dot{H}^{-1})$ satisfying $q(0) = q_0$.

To prove Theorem 1.2, the crucial step is to obtain the $H^2$ bound for the solution $q$ from the above three conserved quantities. This aim is achieved by combining change of variable, interpolation inequalities, and the continuous lemma.

This paper is organized as follows. In Section 2, we use the regularized operator to construct a regularized equation for (1.4) and prove the global well-posedness of smooth solution to this equation. Giving an a priori estimate for the regularized equation and taking the limit argument, Theorem 1.1 is proved in Section 3. Finally, in Section 4, we present the proof of Theorem 1.2.

2.   A regularization of the complex short pulse equation

In this section, we will prove the existence of solution for a regularized problem of the complex short pulse equation. To this end, we introduce the regularized operator $J_\epsilon = (I-\epsilon\partial_{xx})^{-1}$ and consider the following regularized equation

Formally, when $\epsilon\rightarrow0$, Eq (2.1) converges to (1.4). Rigorous justification of this convergence behavior actually gives the proof of Theorem 1.1. Here, we show the global well-poseness for the regularized equation (2.1).

Using the approach of Fourier transform, we can derive the following properties for the operator $J_\epsilon$.

Lemma 2.1. The following two statements hold for $J_\epsilon$.

(i) Suppose $f(x)\in H^k (\mathbb{R})$ with $k\geq 0$. Then,

where $C(\epsilon)$ denotes a constant depending on $\epsilon$.

(ii) If $u(x)\in L^2 (\mathbb{R})$, $v(x)\in L^2 (\mathbb{R})$, then

Proof. Using the definition of the $H^{k+2}$-norm and the property of the Fourier transform, for any given $\epsilon > 0$, one gets

Hence, the estimate (2.3) holds. The proof of (2.2) is similar. For (2.4), we have

The equality (2.5) follows by using the fundamental property of the Fourier transform. □

Taking the basic $L^2$ energy estimate, we can obtain following conservation law.

Lemma 2.2. Suppose that $q^\epsilon\in C([0, T];H^2\cap \dot{H}^{-1})$ is a solution of (2.1). Then, we have

Proof. Note that the above assumption implies $\partial_x^{-1}q^\epsilon\in C([0, T];H^1)$, so $\partial_x^{-1}q^\epsilon$ decays to zero as $|x|\rightarrow \infty$. Multiplying (2.1) by $2\overline{q^\epsilon}$ and integrating the real part over $\mathbb{R}$, we obtain

By Lemma 2.1 $(ii)$, we have

So the $L^2$ conserved quantity is proved. □

From Lemma 2.2 and (2.4), we see that in the linear level the role of $J_\epsilon$ gives $L^\infty$ estimate of $J_\epsilon q^\epsilon$ which is crucial in the global extension argument of Theorem 2.1. Moreover, as we will see later, in the nonlinear level, the appearance of $J_\epsilon$ absorbs the derivative in the nonlinear term which makes the estimate for such terms easier.

Now, we give the global existence result for the regularized problem (2.1).

Theorem 2.1. For any given $\epsilon > 0$ and $q^\epsilon_0 \in H^k\cap \dot{H}^{-1}$ with $k\geq 2$ as an integer, the Eq (2.1)admits a unique solution

satisfying $q^\epsilon(0) = q^\epsilon_0$.

Proof. Our proof is based on the contraction mapping principle and continuation principle of an autonomous ODE on a Banach space. We split the proof into two steps.

1) First, we prove the local well-posedness of the Eq (2.1). The fundamental solution of the linear problem for (2.1)

is

Note that the solution operator $e^{-t\partial_x^{-1}}$ is a norm-preserving map from $H^k$ to $H^k$ in the sense of

By Duhamel's principle, we obtain the integral equation for the nonlinear problem (2.1) satisfying $q^\epsilon(0) = q^\epsilon_0$:

from which we notice that $Q^\epsilon_0 = q^\epsilon_0$.

Define the operator $\Phi$ by

To prove the local well-posedness result, we need to show $\Phi$ maps $H^k$ into $H^k$ and $\Phi$ is locally Lipschitz continuous in $H^k$.

By the triangle inequality, the norm preserving property (2.6) and Lemma 2.1 $(i)$, we deduce

where, for the last inequality, we have used the Banach algebra property (see, e.g., ^[28])

Therefore, this shows that the map $\Phi$ is a closed map of $H^k$ to itself. Moreover, a similar analysis can be used to prove that $\Phi$ is Lipschitz with respect to $q^\epsilon$ and it is contractive if $T_\epsilon = T(\epsilon, q^\epsilon_0) > 0$ is sufficiently small.

Hence, by the contraction mapping principle, we know that there exists a unique solution $q^\epsilon(t, x)\in C([0, T_\epsilon); H^k)$ for some $T_\epsilon > 0$. As $q^\epsilon_0 \in \dot{H}^{-1}$, we can also show $q^\epsilon(t, x)\in C([0, T_\epsilon); H^{-1})$. Indeed, differentiating equation (2.1) gives

which is equivalent to

Thus, we have

which implies

This shows that $q^\epsilon_t(t, x)\in H^{k-2}$ for

Moreover, with similar arguments, we can easily obtain

so $q^\epsilon_t \in C([0, \tau_0];H^{k-2})$. Since $\tau_0$ depends only on the $H^k$ energy norm of $q^\epsilon$, a bootstrap argument shows that the $H^{k-2}$-norm continuity of $q^\epsilon_t$ holds in the whole interval $[0, T_\epsilon)$. This result together with Eq (2.1) also imply that

2) Second, we will show that $T_\epsilon = +\infty$. Assume the maximal existence time $T_\epsilon < +\infty$, from the continuation principle, it suffices for us to obtain an a priori bound for $\|q^\epsilon(\cdot, t)\|_{H^k}$ in the time interval $[0, T_\epsilon)$. From (2.1), we have

Multiplying (2.8) by $2\overline{q^\epsilon_x}$ and integrating the real part over $\mathbb{R}$, we obtain

Applying Cauchy-Schwarz inequality and Lemma 2.1 $(i)$, the nonlinear term is estimated by

where, for the last inequality we have used Lemma 2.2 and (2.4). As a result, we conclude that

which, by Gronwall's inequality, gives us

Taking the partial derivative $\partial_x$ to Eq (2.8) and multiplying both sides by $2\overline{q^\epsilon_{xx}}$, we have

with

We use Hölder's inequality and (2.4) to obtain

For the term $I_3$, we first integrate it by part, then we have

Hence, by (2.11) and Gronwall's inequality, we get

which yields the boundedness of the $H^2$ norm for $q^\epsilon$.

Applying a similar argument as above, we can actually obtain

which implies $T_{\epsilon} = +\infty$. Hence, the proof of Theorem 2.1 is finished. □

We remark that the constant $C(\epsilon)$ that appears in the above proof tends to $\infty$ as $\epsilon\rightarrow 0$.

3.   Local existence and uniqueness

We now present the proof of Theorem 1.1.

Proof of Theorem 1.1. To prove the local existence result of Theorem 1.1, we should drive an a priori estimate for the solution $q^\epsilon$ of the regularized equation. Note that the constant $C$ obtained in (2.10) and (2.12) depends on $\epsilon$, which are not sufficient for our argument. We first regularize the initial data by

where $\rho_\epsilon(x) = \frac{1}{\epsilon}\rho(\frac{x}{\epsilon})$ and $\rho(x)$ is a radial function satisfying

Clearly, we have $q^\epsilon_0(x)\in H^m(\mathbb{R})\cap \dot{H}^{-1}(\mathbb{R})$ for all $m\geq 0$, and

See [29, Section 3.5] for the proof for (3.1). By Theorem 2.1, Eq (2.1) has a unique solution $q^\epsilon\in C([0, +\infty); H^m\cap \dot{H}^{-1})$ equipped with the initial data $q^\epsilon_0(x)$, where the integer $m$ can be taken large enough to proceed all the following differential calculations.

From Lemma 2.2, we have

Next, we will estimate $L^2$ norm of $q^\epsilon_x$. By Lemma 2.1 $(ii)$, the triangle inequality, we obtain from the equality (2.9),

Using Hölder's inequality, Lemma 2.1 $(i)$, and noting that $H^1(\mathbb{R})\hookrightarrow L^\infty(\mathbb{R})$, the first term $\widetilde{I_1}$ is estimated by

For the term $\widetilde{I_2}$, estimating in the same way, there holds

So, we get from these two estimates that

Integrating with respect to $t$ on both sides of (3.3), we obtain that

Now we will estimate the $L^2$ norm of $q^\epsilon_{xx}$. For differentiating equation (2.8) with respect to $x$, we get

Multiplying (3.5) by $2\overline{q^\epsilon_{xx}}$ and taking the real part of the result, we have

Using Lemma 2.1 $(ii)$ and the triangle inequality, we obtain

By Hölder's inequality, Lemma 2.1 $(i)$, the embedding relation $H^1(\mathbb{R})\hookrightarrow L^\infty(\mathbb{R})$, and the Banach algebra property (2.7), the term $\widetilde{I_3}+\widetilde{I_4}$ can be estimated by

For the term $\widetilde{I_5}$, we integrate it by parts to get

Therefore, we concude that

Integrating with respect to $t$ on both sides of (3.6), we obtain

Combining (3.2), (3.4), and (3.7), we get

where we have used (3.1) in the last step. Note that the constant $C$ is independent of $\epsilon$, and by (3.1), one has

Let

Then, there holds

which gives us

Hence, using (3.8), we see that there exists a time $T > 0$ $(T = T(\|q_0\|_{H^2}))$ such that

This shows that the regularized solution $q^\epsilon$ is uniformly bounded in $H^2$, and the argument used in the first step of Theorem 2.1 also yields the uniform bound of $q^\epsilon_t$ in $L^2$.

Moreover, we can show that the solution family $q^\epsilon$ (also $q^\epsilon_t$) forms a Cauchy sequence in $C([0, T];L^2)$. In fact, for $\epsilon > \epsilon' > 0$, taking energy estimate of the equation

at $L^2$ level yields that

where

Using Plancherel's theorem and (3.10), we have

Therefore,

To estimate $K_2$, we rewrite it as

with

Using the uniform bound (3.10) and the same treatment for $J_\epsilon-J_{\epsilon'}$ as above, we can get

Similarly, the term $K_{23}$ is estimated by

Then, integrating by part gives us

Combining these estimates gives us

By interpolation, we also have

for any $0\leq k' < 2$. A similar strategy can be applied to show the Cauchy property of the sequence $q^\epsilon_t$ in $C([0, T];L^2)$, that is

and further details are omitted.

Now we can prove the existence part of Theorem 1.1. Indeed, from (3.12), (3.13), and the equation (2.1), applying the standard limit argument, we see that there exists $q\in C([0, T];H^{k'}\cap \dot{H}^{-1})\cap C_W([0, T];H^2)$ satisfying Eq (1.4) with $k' < 2$, here $C_W([0, T];H^2)$ denotes the continuity on $[0, T]$ with values in the weak topology of $H^2$. Furthermore, we can show that $q$ is also continuous in the strong topology of $H^2$. To see this result, we rewrite (3.9) in the form

By (3.8) and the weak convergence property of $q^\epsilon(t)$ in $H^2$, we obtain

which implies that

On the other hand, the fact $q\in C_W([0, T];H^2)$ gives

Hence, the strong continuity of $q$ at $t = 0$ is proved. This argument also yields the continuity of $q$ in $H^2$ at any time. Then, we have $q\in C([0, T];H^2)$.

Finally, it remains to prove the uniqueness. In fact, if $q$ and $\tilde{q}$ both satisfy (1.4) with the same initial data, then $q-\tilde{q}$ satisfies

Multiplying (3.14) by $2\overline{q-\tilde{q}}$ and integrating the real part over $\mathbb{R}$, we get

Then, applying integration by parts, Hölder's inequality, and the Banach algebra property (2.7), we obtain

where we have used the fact that $q, \tilde{q}\in H^2$ in the last inequality. Since $q$ and $\tilde{q}$ satisfy the same initial data, the uniqueness follows from Gronwall's inequality.

□

4.   Global well-posedness

In this section, we will prove the global well-posedness of the complex short pulse equation (1.3), namely, Theorem 1.2. The proof is based only on the energy analysis. To prove this result, we need to control the $H^2$ norm of $q(t)$ by a $t$-independent constant. This constant will be found from the values of the conserved quantities of (1.4).

Lemma 4.1. Let $q(t, x)\in C([0, T);H^2\cap \dot{H}^{-1})$ be the solution of (1.4) obtained in Theorem 2.1. Then, the following quantities are conserved on $[0, T)$:

Proof. For the sake of simplicity, $q$ is assumed smooth in this proof and rigorous limit argument is omitted. Multiplying (1.4) by $2\overline{q}$, and taking the real part, we derive the first balance equation

where $\partial^{-1}_xq = -q_t-\frac{1}{2}|q|^2q_x$.

Multiplying (1.3) by $\frac{2\overline{q_x}}{\sqrt{1+|q_x|^2}}$, and taking the real part, we derive the second balance equation

Integrating (4.1) and (4.2) over $x$ in $\mathbb{R}$, we obtain the conversations of $\mathcal{H}_1(t)$ and $\mathcal{H}_2(t)$.

Differentiating equation (1.3) with respect to $x$, we get

Multiplying (4.3) by $\frac{2\overline{q_{xx}}}{(1+|q_x|^2)^\frac{5}{2}}$, taking the real part, and noting that

we obtain

From Eq (1.3), we have

Thus, we deduce

By a direct computation, we get

From Eqs (1.3) and (4.3), we have

and

Note that

So, we deduce

Combining (4.4) and (4.5), we conclude

where, we have used the following equality

Integrating (4.6) over $x$ in $\mathbb{R}$, we obtain the conversation of $\mathcal{H}_3(t)$. □

To control the $H^2$ norm of $q(t)$, we also need the following two results, which can be found in ^[30] and [31, Lemma 2.1.3], respectively.

Lemma 4.2. Let $u$ belongs to $L^b$ in $\mathbb{R}^n$ and its derivatives of order $m$, $D^mu$, belongs to $L^r$, $1\leq b, r\leq\infty$. For the derivatives $D^ju$, $0\leq j < m$, the following inequalities hold

where

for all $p$ in the interval $\frac{j}{m}\leq p\leq 1$ (The constant depending only on $n, m, j, b, r, p$).

Lemma 4.3. Let $f(x)$ be a nonnegative continuous function on $\mathbb{R}^+$ satisfying

If $a$ and $b$ further satisfy

and $f(0)\leq a$, then $f(x)$ is bounded on $\mathbb{R}^+$.

Proof of Theorem 1.2. As shown in Theorem 2.1, the $\dot{H}^{-1}$ norm of $q(t, x)$ is essentially controlled by its $H^2$ energy norm. To estimate $\|q(t)\|_{H^2}$, noting that $\mathcal{H}_1(t) = \|q(t)\|^2_{L^2}$ is a constant, it suffices for us to estimate $\|q_x(t)\|_{H^1}$. To this end, we introduce a variable

that is,

When $|Q| < 1$, we can expand $q_x$ in the Taylor series

By the Banach algebra property (2.7) and the triangle inequality, we have

where the equality in (4.7) holds under the condition $\|Q\|_{H^1} < 1$.

Next, we shall give an upper bound for $\|Q\|_{H^1}$. On one hand,

On the other hand, noting that

and

by a direct computation, we get

Then,

By Lemma 4.2, we have

Note that the condition of Lemma 4.3 holds with the smallness assumption on $\mathcal{H}_1$ and $\mathcal{H}_3$. Hence, applying Lemma 4.3 implies that $\|q_x\|_{L^\infty}$ has an upper bound $C(\mathcal{H}_1, \mathcal{H}_3)$, provided that $\mathcal{H}_1(0)$ and $\mathcal{H}_3(0)$ are sufficiently small. So from (4.8)–(4.9), we see that the norm

is bounded from above by a constant depending on $\mathcal{H}_1(0)$, $\mathcal{H}_2(0)$ and $\mathcal{H}_3(0)$, that is,

Combining (4.7) and (4.10), the following holds

which results in the $t$-independent bound of the norm $\|q\|_{H^2}$

This bound allows us to apply bootstrap argument, and obtain global existence of solution to the Eq (1.4). This completes the proof of Theorem 1.2. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the anonymous referees for valuable comments and suggestions. This work is supported by Zhejiang Provincial Natural Science Foundation of China (Grant No. LY23A010006), and the NSFC Grants 11771183, 11971503.

Conflict of interest

The authors declare there is no conflict of interest.